#501324

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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

690 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)(8,10)(9,12)(13,14,18,22,31,37,43,42,17,21,30,36,19,23,33,39)(15,24,25,40,35,29,38,32,20,16,27,28,26,34,41,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,17,20)(14,16,21,24)(18,25,30,27)(19,26,31,35)(22,28,36,40)(23,29,37,34)(32,42,44,39)(33,41,43,38);(2,4)(13,16,17,24)(14,20,21,15)(18,32,30,44)(19,34,31,29)(22,38,36,41)(23,26,37,35)(25,39,27,42)(28,43,40,33);(6,10)(9,11)(14,21)(15,20)(22,36)(23,37)(25,27)(26,35)(38,41)(39,42)

Polycyclic presentation:

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

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, 408520);