#6605666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285415
maximal normal subgroup of	#72285415

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512862);