#310514

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 217710);