#502861

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
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

860 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)(13,14,18,23)(15,20,26,38)(16,29,31,41)(17,34,36,42)(19,37,35,24)(21,39,28,32)(22,40,25,30)(27,33,43,44);(1,2)(3,4)(6,9)(10,11)(13,15)(14,20)(16,22)(17,27)(18,26)(19,28)(21,35)(23,38)(24,39)(25,31)(29,40)(30,41)(32,37)(33,34)(36,43)(42,44);(2,4)(13,16,30,20,19,33,36,39,18,31,40,38,35,44,17,32)(14,21,34,27,24,15,25,41,23,28,42,43,37,26,22,29);(5,7,8,12)(6,10,9,11)(15,26)(16,31)(20,38)(21,28)(27,43)(29,41)(32,39)(33,44)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=e,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, 410057);