#6474766

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(22,25)(24,27)(28,32)(29,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,24)(20,22)(21,23)(25,30)(26,29)(27,32)(28,31);(11,15)(13,16)(17,19,25,32,21,28,22,26)(18,24,30,27,23,31,20,29);(11,15)(13,16)(17,20)(18,25)(19,27)(21,30)(22,23)(24,26)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,23)(19,28)(20,30)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6381962);