#313195

order	512
nilpotency class	3
isoclinism	isoclinic class of #264427
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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 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)(4,6)(5,8)(10,13)(14,16)(17,18)(19,27)(20,23)(21,30)(22,26)(24,28)(25,31)(29,32);(1,3,4,8)(2,5,6,7)(10,14)(12,15)(13,16)(17,19,21,20,22,28,31,29)(18,23,25,27,26,32,30,24);(2,6)(5,7)(9,10)(11,13)(12,16)(14,15)(19,28)(20,29)(23,32)(24,27);(2,6)(5,7)(17,20,21,28,22,29,31,19)(18,24,25,23,26,27,30,32)

Polycyclic presentation:

2^9|b^2=e,d^2=efi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,d^c=dh,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, 220391);