#6720070

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1415 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)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,29,38)(20,24,31,40)(21,25,35,44)(27,36,45,47)(28,37,33,42)(30,39,46,48)(32,41,34,43);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(17,19)(18,23)(20,30)(21,33)(22,29)(24,39)(25,42)(26,38)(27,32)(28,35)(31,46)(34,45)(36,41)(37,44)(40,48)(43,47);(2,7)(6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,43)(26,40)(28,46)(29,45)(30,33)(32,35)(37,48)(38,47)(39,42)(41,44);(9,12,14,13)(10,15,11,16)(20,31)(24,40)(27,45)(30,46)(32,34)(36,47)(39,48)(41,43);(1,4,5,3)(2,8,7,6)(9,13,14,12)(10,15,11,16)(19,28)(21,35)(23,37)(25,44)(27,46)(29,33)(30,45)(32,34)(36,48)(38,42)(39,47)(41,43)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=f,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6627266);