#306121

order	512
nilpotency class	3
isoclinism	isoclinic class of #264427
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

608 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=f,d^2=e,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, 213317);