#6412395

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

1625 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)(17,18,22,26,21,25,34,41)(19,23,29,37,28,36,43,47)(20,30,33,44,32,24,38,40)(27,39,42,48,31,35,45,46);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,27)(21,28)(22,29)(24,35)(25,36)(26,37)(30,39)(31,32)(33,42)(34,43)(38,45)(40,46)(41,47)(44,48);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,20,22,33,21,32,34,38)(18,24,26,40,25,30,41,44)(19,27,29,42,28,31,43,45)(23,35,37,46,36,39,47,48);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,31)(21,28)(22,29)(24,39)(25,36)(26,37)(27,32)(30,35)(33,45)(34,43)(38,42)(40,48)(41,47)(44,46);(9,12,13,11)(10,15,14,16)(19,28)(23,36)(27,31)(29,43)(35,39)(37,47)(42,45)(46,48)

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=dgi,e^a=ef,c^b=cgi,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, 6319591);