#307490

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
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)(3,7)(4,5)(6,8)(9,10,14,15)(11,16,12,13)(17,18,21,25)(19,26,29,23)(20,30,22,24)(27,32,28,31);(2,5)(6,7)(9,11,13,10,14,12,16,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(9,12,13,15,14,11,16,10)(19,27)(20,22)(23,31)(24,30)(26,32)(28,29);(1,3,4,8)(2,6,5,7)(9,11,13,10,14,12,16,15)(17,20,21,22)(18,24,25,30)(19,28,29,27)(23,31,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 214686);