#491674

order	512
nilpotency class	4
isoclinism	isoclinic class of #491644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33771	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,27)(20,23)(21,34)(22,35)(24,30)(25,28)(26,33)(29,39)(31,46)(32,47)(36,44)(37,43)(38,48)(40,42)(41,45);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(18,23)(24,36)(25,37)(26,35)(27,39)(34,47)(38,40)(41,46);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,19,28,42,22,31,32,44,20,29,43,48,33,45,21,30)(18,24,34,41,26,38,37,39,23,36,47,46,35,40,25,27);(3,10)(8,15)(11,14)(12,16)

Polycyclic presentation:

2^9|b^2=egi,c^2=fh,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^c=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398870);