#503548

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

755 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=efh,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di

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, 410744);