#6469930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(4,8)(5,6)(17,18,23,24,22,28,21,27)(19,25,30,31,20,26,29,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(21,23)(25,31)(26,32)(27,28)(29,30);(2,6)(7,8)(10,13)(14,16)(17,19)(18,25)(20,22)(21,29)(23,30)(24,31)(26,28)(27,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,19)(18,26,28,25)(21,30,23,29)(24,32,27,31);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(19,20)(25,26)(29,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377126);