#470290

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377486);