#7032388

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=f,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=ehi,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, 6939584);