#382876

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,d^2=f,b^a=be,c^a=cf,d^a=di,c^b=cg,h^b=hi,d^c=de,g^c=gi

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, 290072);