#6718603

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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)(12,13)(15,16)(17,18,21,25)(19,27,31,23)(20,24,33,40)(22,26,35,42)(28,38,30,36)(29,39,32,37)(34,41,45,48)(43,47,44,46);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(17,19,21,31)(18,23,25,27)(20,32,33,29)(22,28,35,30)(24,39,40,37)(26,36,42,38)(34,44,45,43)(41,47,48,46);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(19,28)(20,33)(22,35)(23,36)(24,40)(26,42)(27,38)(29,44)(30,31)(32,43)(37,47)(39,46);(10,13)(14,16)(17,20,21,33)(18,24,25,40)(19,29,31,32)(22,34,35,45)(23,37,27,39)(26,41,42,48)(28,43,30,44)(36,46,38,47);(1,4,5,3)(2,8,7,6)(19,30)(22,35)(23,38)(26,42)(27,36)(28,31)(29,44)(32,43)(34,45)(37,47)(39,46)(41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,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, 6625799);