#6359541

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#61154	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,g^c=gi,f^d=fh

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, 6266737);