#309626

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
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

596 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,16,15)(17,18,21,24)(19,25,27,31)(20,28,30,23)(22,29,32,26);(9,11)(10,14)(12,16)(13,15)(17,19,21,27)(18,22,24,32)(20,29,30,26)(23,31,28,25);(2,5)(6,8)(10,13)(11,15)(14,16);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(18,23)(20,30)(22,31)(24,28)(25,32)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 216822);