#6720096

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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)(17,18,21,25)(19,27,30,23)(20,24,33,40)(22,26,35,42)(28,45,46,37)(29,38,31,43)(32,48,47,39)(34,41,36,44);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(17,19)(18,23)(20,32)(21,30)(22,29)(24,39)(25,27)(26,43)(28,34)(31,35)(33,47)(36,46)(37,41)(38,42)(40,48)(44,45);(2,7)(6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,24)(19,28)(21,33)(22,36)(23,37)(25,40)(26,44)(27,45)(29,32)(30,46)(31,47)(34,35)(38,48)(39,43)(41,42);(9,12,14,13)(10,15,11,16)(18,25)(20,33)(23,27)(26,42)(28,46)(32,47)(34,36)(38,43);(1,4,5,3)(2,8,7,6)(9,13,14,12)(10,15,11,16)(18,25)(19,29)(22,35)(23,38)(24,40)(27,43)(28,32)(30,31)(34,36)(37,48)(39,45)(46,47)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6627292);