#7093590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1372 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,30,21,27)(20,26,36,44)(22,28,23,38)(25,32,40,33)(31,45,34,41)(35,42,37,43)(39,47,48,46);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33,24,21,22,32)(18,25,28,30,29,40,38,27)(20,31,35,46,36,34,37,47)(26,39,43,45,44,48,42,41);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,36)(18,26,29,44)(19,31,21,34)(22,37,23,35)(25,39,40,48)(27,41,30,45)(28,43,38,42)(32,47,33,46);(11,16)(13,15)(17,21,24,19)(18,27,29,30)(20,34,36,31)(22,32,23,33)(25,28,40,38)(26,41,44,45)(35,46,37,47)(39,43,48,42);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,32,21,33)(20,35,36,37)(25,30,40,27)(26,42,44,43)(31,46,34,47)(39,41,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7000786);