#213889

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

842 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,13,16,8)(4,18,20,9)(5,22,25,10)(6,11)(7,12)(14,38,41,31)(15,28)(17,29,43,30)(19,48,51,32)(21,47,50,33)(23,34)(24,35)(26,56,58,36)(27,40,45,42)(37,44)(39,61,49,60)(46,53,55,52)(54,57)(59,63,62,64);(1,3,7,17)(2,8,12,30)(4,14,20,42)(5,23,26,15)(6,16,24,43)(9,31,18,45)(10,34,36,28)(11,13,35,29)(19,49,53,39)(21,40,50,41)(22,54,56,37)(25,57,58,44)(27,47,38,33)(32,61,46,60)(48,59,55,62)(51,63,52,64);(1,4,6,21)(2,9,11,33)(3,14,16,40)(5,19,25,52)(7,20,24,50)(8,27,13,31)(10,32,22,55)(12,18,35,47)(15,39,44,63)(17,42,43,41)(23,49,57,64)(26,53,58,51)(28,59,37,60)(29,45,30,38)(34,62,54,61)(36,46,56,48);(1,5,24,58)(2,10,35,56)(3,15,43,57)(4,19,50,51)(6,25,7,26)(8,28,29,54)(9,32,47,48)(11,22,12,36)(13,37,30,34)(14,39,41,64)(16,44,17,23)(18,46,33,55)(20,53,21,52)(27,59,45,61)(31,60,38,62)(40,63,42,49)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268306
maximal normal subgroup of	#72268306
maximal quotient of	#72268315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121085);