#6720055

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6627251);