#7409869

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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)(9,10,13,12)(11,14,15,16)(17,18)(19,28)(20,31)(21,26)(22,27)(23,24)(25,29)(30,41)(32,44)(33,42)(34,40)(35,37)(36,39)(38,43)(45,48)(46,47);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19,22,23)(18,24,27,28)(20,29,33,36)(21,30,34,35)(25,31,39,42)(26,37,40,41)(32,43,45,46)(38,44,47,48);(10,12)(11,15)(17,20,22,33)(18,25,27,39)(19,29,23,36)(21,32,34,45)(24,31,28,42)(26,38,40,47)(30,43,35,46)(37,44,41,48);(1,3,5,4)(2,6,7,8)(9,12,13,10)(11,14,15,16)(17,21,22,34)(18,26,27,40)(19,30,23,35)(20,32,33,45)(24,37,28,41)(25,38,39,47)(29,43,36,46)(31,44,42,48);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(18,27)(21,34)(24,28)(25,39)(30,35)(31,42)(32,45)(43,46)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^2=fh,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=eh,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, 7317065);