#498724

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂×#121	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=dei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405920);