#6412392

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	#36687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,21,25,22,26,33,40)(19,23,29,36,28,35,42,46)(20,24,31,38,32,39,44,48)(27,34,41,45,30,37,43,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,27)(21,29)(22,28)(24,34)(25,36)(26,35)(30,32)(31,41)(33,42)(37,39)(38,45)(40,46)(43,44)(47,48);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,20,21,31,22,32,33,44)(18,24,25,38,26,39,40,48)(19,27,29,41,28,30,42,43)(23,34,36,45,35,37,46,47);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,37)(25,36)(26,35)(27,32)(31,43)(33,42)(34,39)(38,47)(40,46)(41,44)(45,48);(9,12,13,11)(10,15,14,16)(19,28)(23,35)(27,30)(29,42)(34,37)(36,46)(41,43)(45,47)

Polycyclic presentation:

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

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, 6319588);