#7027588

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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

1410 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,25)(21,23)(22,34)(24,30)(26,33)(28,40)(29,39)(31,36)(32,45)(35,43)(37,42)(38,44)(41,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(24,35)(26,34)(27,39)(28,41)(32,44)(42,48);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,29,33,43)(18,24,26,27,23,35,34,39)(20,28,32,42,31,41,44,48)(25,37,38,40,36,46,45,47);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,28)(21,31)(22,32)(23,36)(24,37)(26,38)(27,40)(29,41)(30,42)(33,44)(34,45)(35,46)(39,47)(43,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(20,31)(25,36)(28,41)(32,44)(37,46)(38,45)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,e^2=fi,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^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6934784);