#472596

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 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)(6,9)(10,12)(13,14,16,18)(15,19,20,17)(21,22,25,28)(23,26,30,34)(24,27,32,36)(29,33,31,35);(1,2,4,3)(5,6,8,9)(7,10,11,12)(21,23,24,31,25,30,32,29)(22,26,27,35,28,34,36,33);(2,3)(7,11)(10,12)(13,15,16,20)(14,17,18,19)(23,29)(24,32)(26,33)(27,36)(30,31)(34,35);(1,3,4,2)(5,7,8,11)(6,10,9,12)(23,30)(26,34)(29,31)(33,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,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, 379792);