#508432

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

559 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=e,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, 415628);