#346251

order	512
nilpotency class	3
isoclinism	isoclinic class of #346224
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

456 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14,17,8)(4,19,40,35)(5,24,25,10)(6,11)(7,13)(9,22,34,23)(12,27)(15,39,42,48)(16,44,63,36)(18,31,47,30)(20,28,43,32)(21,51,55,57)(26,38,58,37)(29,45,56,46)(33,53,49,54)(41,60)(50,64)(52,61)(59,62);(1,3,7,18)(2,8,13,30)(4,20,22,43)(5,16,25,46)(6,17,27,47)(9,32,35,28)(10,36,24,56)(11,14,12,31)(15,40,42,23)(19,48,34,39)(21,52,53,50)(26,45,58,63)(29,38,44,37)(33,61,57,64)(41,55,62,54)(49,59,51,60);(1,4)(2,9,13,35)(3,15,17,20)(5,21)(6,23)(7,22)(8,28,31,48)(10,33,24,57)(11,19,12,34)(14,39,30,32)(16,41,45,50)(18,42,47,43)(25,53)(26,55)(27,40)(29,59,56,64)(36,61,44,60)(37,49,38,51)(46,62,63,52)(54,58);(1,5,6,26)(2,10,11,37)(3,16,17,45)(4,21,23,55)(7,25,27,58)(8,29,14,36)(9,33,19,49)(12,38,13,24)(15,41,20,50)(18,46,47,63)(22,53,40,54)(28,59,39,61)(30,44,31,56)(32,60,48,64)(34,51,35,57)(42,62,43,52)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ef,d^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=de,f^b=fh,e^c=ei,f^c=fi,g^c=gh,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284584, #72284636, #72284721, …
maximal normal subgroup of	#72284584, #72284636, #72284721, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253447);