#6378568

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#36692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285764);