#6469914

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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1519 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)(17,18,21,25,22,26,30,23)(19,24,27,32,20,28,29,31);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(21,30)(24,31)(25,26)(27,29)(28,32);(2,6)(7,8)(10,13)(14,16)(17,19,22,20)(18,24,26,28)(21,27,30,29)(23,31,25,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,19)(18,24,26,28)(21,29,30,27)(23,31,25,32);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(19,20)(24,28)(27,29)(31,32)

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,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, 6377110);