#6630860

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629007
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,23)(21,31)(22,26)(24,29)(25,28)(30,32);(2,5)(6,8)(17,19,28,30,22,29,21,20)(18,23,31,24,26,32,25,27);(9,11,12,16)(10,13,14,15);(1,3,4,7)(2,6,5,8)(17,20,21,29,22,30,28,19)(18,24,25,23,26,27,31,32);(19,29)(20,30)(23,32)(24,27)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,e^b=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224688, #72224725, #72224758
maximal normal subgroup of	#72224688, #72224725, #72224758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538056);