#299500

order	512
nilpotency class	3
isoclinism	isoclinic class of #299469
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	#18620	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1007 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=f,d^2=ei,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,d^b=di,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206696);