#7010302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1352 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6917498);