#6811935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3180	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22)(23,33)(24,31)(25,36)(26,32)(27,30)(28,35)(29,34);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,26,35)(22,28,32,33)(24,25,34,27)(29,36,31,30);(2,4)(21,24,27,23,26,34,25,35)(22,29,30,28,32,31,36,33);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,27)(22,30,32,36)(23,34,35,24)(28,29,33,31);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,25,35,26,34,27,23)(22,31,36,28,32,29,30,33)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256965, #72257027, #72257047
maximal normal subgroup of	#72256965, #72257027, #72257047
maximal quotient of	#72264873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719131);