#499976

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3339	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

963 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,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, 407172);