#502862

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 410058);