#299261

order	512
nilpotency class	3
isoclinism	isoclinic class of #299253
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60716	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

917 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,g^b=gh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206457);