#6495052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,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, 6402248);