#300056

order	512
nilpotency class	3
isoclinism	isoclinic class of #300013
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 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)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18,24,27)(19,28,31,21)(20,22,32,23)(25,26,29,30);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(17,19,23,26,24,31,22,30)(18,25,32,21,27,29,20,28);(17,20,22,27,24,32,23,18)(19,29,30,21,31,25,26,28);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(17,21,22,25,24,28,23,29)(18,26,20,19,27,30,32,31)

Polycyclic presentation:

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

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, 207252);