#7090618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,33,44)(20,35,32,25)(21,26,31,46)(22,38,30,27)(24,40,43,34)(36,42,47,48)(37,41,39,45);(9,10)(11,15)(12,14)(13,16)(17,19,30,40,23,33,22,34)(18,24,38,44,28,43,27,29)(20,31,47,37,32,21,36,39)(25,41,48,46,35,45,42,26);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,32)(18,25,28,35)(19,31,33,21)(22,36,30,47)(24,41,43,45)(26,44,46,29)(27,42,38,48)(34,39,40,37);(11,16)(13,15)(17,21,23,31)(18,26,28,46)(19,32,33,20)(22,39,30,37)(24,42,43,48)(25,44,35,29)(27,41,38,45)(34,36,40,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,33)(20,32)(24,43)(25,35)(29,44)(34,40)(36,47)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997814);