#6474740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	#24792	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 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)(20,21)(24,26)(25,31)(27,32)(28,30);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,24)(23,26)(25,27)(31,32);(11,15)(13,16)(17,18,20,26,22,24,21,23)(19,27,30,31,29,25,28,32);(11,15)(13,16)(17,19)(18,25)(20,30)(21,28)(22,29)(23,31)(24,27)(26,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,22,21)(18,26,24,23)(19,28,29,30)(25,31,27,32)

Polycyclic presentation:

2^9|a^2=f,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,d^c=di,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, 6381936);