#7198506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(10,14)(12,15)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,32)(28,40)(29,37)(31,44)(33,39)(34,41)(35,42)(38,45)(43,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,16,15)(17,19,23,22)(18,24,27,36)(20,28,33,32)(21,29,35,34)(25,30,40,39)(26,37,42,41)(31,43,46,45)(38,44,47,48);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,20,22,32,23,33,19,28)(18,25,24,30,27,40,36,39)(21,31,34,45,35,46,29,43)(26,38,37,44,42,47,41,48);(3,8)(6,7)(10,14)(11,16)(17,21,23,35)(18,26,27,42)(19,29,22,34)(20,31,33,46)(24,37,36,41)(25,38,40,47)(28,43,32,45)(30,44,39,48);(9,12,13,15)(10,11,14,16)(17,19,23,22)(18,24,27,36)(20,32,33,28)(21,34,35,29)(25,39,40,30)(26,41,42,37)(31,43,46,45)(38,44,47,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fgh,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105702);