#6614823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,23,26)(19,27,28,30)(20,29,31,24)(21,25,22,32);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,24,23,28,21,29)(18,20,25,27,26,31,32,30);(1,3,5,4)(2,7,6,8)(11,16)(13,14)(17,20,23,31)(18,24,26,29)(19,25,28,32)(21,30,22,27);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,22)(18,25,26,32)(19,24,28,29)(20,30,31,27);(11,16)(13,14)(19,28)(20,31)(24,29)(27,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=g,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

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, 6522019);