#7086553

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1381 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 6993749);