#6495198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

1574 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402394);