#415168

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,19,20,17)(21,22,25,27)(23,26,31,34)(24,28,32,36)(29,33,30,35);(1,2,4,3)(6,9)(10,12)(21,23)(22,26)(24,29)(25,31)(27,34)(28,33)(30,32)(35,36);(1,3,4,2)(5,7,8,11)(6,10,9,12)(23,29)(24,32)(26,33)(28,36)(30,31)(34,35);(13,15,16,20)(14,17,18,19)(22,27)(26,34)(28,36)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322364);