#6458447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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)(9,10)(11,14)(12,16)(13,15)(17,18,23,20,22,26,21,25)(19,24,30,28,29,31,27,32);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(18,20)(19,27)(21,23)(25,26)(28,32)(29,30);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,19,21,27,22,29,23,30)(18,24,25,32,26,31,20,28);(1,4,5,3)(2,8,6,7)(17,20,21,18,22,25,23,26)(19,28,27,24,29,32,30,31);(19,29)(24,31)(27,30)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365643);