#508282

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

757 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,28)(17,22)(18,21)(19,23)(24,26)(25,36)(27,37)(29,40)(30,32)(31,38)(33,39)(34,35)(41,44)(42,43);(6,9)(10,12)(13,14)(15,24)(16,21)(17,30)(18,28)(19,23)(20,26)(22,32)(25,41)(27,42)(29,39)(31,35)(33,40)(34,38)(36,44)(37,43);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,27)(14,20,23,37)(16,25,18,34)(17,31,33,44)(21,35,28,36)(22,38,39,41)(24,40,42,30)(26,29,43,32);(7,11)(10,12)(13,16,19,18)(14,21,23,28)(15,25,27,34)(17,29,33,32)(20,35,37,36)(22,30,39,40)(24,41,42,38)(26,31,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415478);