#7046709

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1306 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)(10,11)(12,16)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,31)(28,40)(29,37)(32,43)(33,38)(34,41)(35,46)(39,44)(42,48)(45,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,15)(14,16)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,29,34,35)(25,38,40,30)(26,37,41,46)(32,44,45,42)(39,47,48,43);(9,11,13,10)(12,15,16,14)(17,20,22,31,23,33,19,28)(18,25,24,30,27,40,36,38)(21,32,35,44,34,45,29,42)(26,39,37,43,41,48,46,47);(3,8)(6,7)(10,11)(14,15)(17,21,23,34)(18,26,27,41)(19,29,22,35)(20,32,33,45)(24,37,36,46)(25,39,40,48)(28,42,31,44)(30,43,38,47);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(18,27)(20,33)(21,34)(24,36)(28,31)(29,35)(39,48)(43,47)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 6953905);