#508924

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

726 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)(7,10)(8,9)(11,12)(13,14,18,24,16,22,31,39,17,23,32,40,19,25,33,21)(15,26,29,42,30,36,44,38,27,20,34,37,28,41,43,35);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,30)(17,27)(18,29)(19,28)(21,38)(22,41)(23,26)(24,37)(25,36)(31,44)(32,34)(33,43)(35,39)(40,42);(2,4)(5,7)(6,10)(8,11)(9,12)(14,21)(15,27)(16,19)(18,33)(20,35)(22,40)(23,39)(24,25)(26,38)(29,44)(31,32)(34,43)(36,42)(37,41);(7,11)(10,12)(13,16,17,19)(14,22,23,25)(15,28,27,30)(18,31,32,33)(20,36,26,41)(21,24,39,40)(29,43,34,44)(35,37,38,42)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fh,h^c=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, 416120);