#6456212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	#34976	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1718 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(2,6)(7,8)(17,19)(18,22)(20,25)(21,27)(23,29)(24,31)(26,28)(30,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(1,4,5,3)(2,8,6,7)(19,26)(20,28)(22,30)(23,32)(25,27)(29,31);(9,11,12,15)(10,14,13,16)(19,27)(22,31)(25,26)(29,30)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363408);