#473790

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1016 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)(13,14,16,17)(15,19,20,18)(21,22,24,27)(23,29,30,26)(25,28,32,35)(31,36,33,34);(1,2,4,3)(5,6,8,9)(7,10,11,12)(14,17)(18,19)(21,23,25,31,24,30,32,33)(22,26,28,34,27,29,35,36);(2,3)(7,11)(10,12)(13,15,16,20)(14,18,17,19)(21,23)(22,26)(24,30)(25,33)(27,29)(28,36)(31,32)(34,35);(1,3,4,2)(5,7,8,11)(6,10,9,12)(15,20)(18,19)(23,30)(26,29)(31,33)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=efgh,c^2=fi,d^2=ef,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, 380986);