#456168

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,22)(17,21)(18,24)(19,25)(20,35)(23,33)(27,43)(28,37)(29,42)(30,39)(31,40)(32,36)(34,41)(38,44);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,18,17)(14,21,24,26)(16,29,30,44)(19,33,20,28)(22,38,39,42)(23,25,37,35)(27,34,32,31)(36,41,43,40);(2,3)(6,10)(9,12)(13,16,18,30)(14,22,24,39)(15,27,17,32)(19,34,20,31)(21,36,26,43)(23,42,37,38)(25,41,35,40)(28,44,33,29);(1,3,4,2)(13,17,18,15)(14,23,24,37)(16,27,30,32)(19,33,20,28)(21,25,26,35)(22,38,39,42)(29,34,44,31)(36,40,43,41)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363364);