#508441

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

580 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)(13,14,19,33,29,41,31,43,16,22,17,23,18,24,36,37)(15,25,34,38,32,39,30,21,26,20,27,44,28,42,35,40);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,34)(21,37)(22,25)(23,38)(24,39)(29,32)(30,31)(33,44)(35,36)(40,43)(41,42);(2,4)(7,11)(10,12)(13,15)(14,21)(16,26)(17,30)(18,32)(19,35)(20,37)(22,40)(23,42)(24,44)(25,43)(27,31)(28,29)(33,39)(34,36)(38,41);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(23,33)(24,41)(27,34)(28,32)(30,35)(37,43)

Polycyclic presentation:

2^9|a^2=ef,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 415637);