#6408527

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di

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, 6315723);