#6475511

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,21,25,22)(20,24,30,32)(27,31,28,29);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,18,23,25,21,26)(20,28,29,32,30,27,31,24);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20)(18,24)(19,27)(21,31)(22,29)(23,30)(25,28)(26,32);(11,15)(13,16)(17,20)(18,24)(19,27)(21,31)(22,29)(23,30)(25,28)(26,32);(10,14)(13,16)(17,21,23,22)(18,19,26,25)(20,29,30,31)(24,28,32,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=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, 6382707);