#6474754

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(17,18,23,25)(19,22,27,21)(20,24,28,32)(26,30,29,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(20,28)(24,32)(26,29)(30,31);(11,15)(13,16)(17,19,22,25,23,27,21,18)(20,29,30,24,28,26,31,32);(11,15)(13,16)(17,20)(18,24)(19,26)(21,31)(22,30)(23,28)(25,32)(27,29);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,22)(18,19,25,27)(20,30,28,31)(24,29,32,26)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,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, 6381950);