#7307881

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7215077);