#6495492

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,24,32,23,25,22,26)(19,27,31,20,29,30,28,21);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,20,25,21)(22,31,24,28)(26,30,32,27);(2,5)(6,8)(10,13)(14,16)(17,20)(18,19)(21,23)(22,27)(24,30)(25,29)(26,28)(31,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,23,20)(18,19,25,29)(22,30,24,27)(26,28,32,31);(9,11,12,15)(10,14,13,16)(18,25)(20,21)(26,32)(27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402688);