#498698

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=ef,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405894);