#245301

order	512
nilpotency class	3
isoclinism	isoclinic class of #245282
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1102 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=hi,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268306
maximal normal subgroup of	#72268306
maximal quotient of	#72268317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152497);