#6371109

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^c=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278305);