#6472866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6380062);