#506222

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

746 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)(13,14,18,23)(15,24,25,41)(16,21,19,33)(17,29,31,43)(20,35,36,27)(22,28,40,32)(26,42,34,39)(30,37,44,38);(1,2,3,4)(5,7,8,12)(6,9,10,11)(15,25)(20,36)(24,41)(26,34)(27,35)(30,44)(37,38)(39,42);(2,4)(13,15)(14,20)(16,26)(17,30)(18,25)(19,34)(21,37)(22,39)(23,36)(24,29)(27,32)(28,35)(31,44)(33,38)(40,42)(41,43);(6,10)(9,11)(13,16,18,19)(14,21,23,33)(15,26,25,34)(17,28,31,32)(20,37,36,38)(22,29,40,43)(24,42,41,39)(27,30,35,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413418);