#6345982

order	512
nilpotency class	3
isoclinism	isoclinic class of #62295
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61435	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1529 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=chi,f^a=fh,c^b=ci,f^b=fh,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253178);