#427499

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1345 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:

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

Polycyclic presentation:

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

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, 334695);