#335696

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 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)(3,4)(5,7)(6,8)(17,18,22,26)(19,23,21,24)(20,28,29,31)(25,30,32,27);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(14,16)(17,19,22,21)(18,23,26,24)(20,27,29,30)(25,31,32,28);(1,4,5,6)(2,3,7,8)(9,10,12,14)(11,16,15,13)(18,24)(19,21)(23,26)(25,28)(27,30)(31,32);(2,7)(3,8)(17,20,22,29)(18,25,26,32)(19,27,21,30)(23,31,24,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ei,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242892);