#455939

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1178 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=i,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, 363135);