#7018712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,24)(19,25,31,39,29,41,36,37)(20,26,32,43,35,44,34,38)(30,40,46,48,33,42,45,47);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,32)(21,22)(25,39)(27,28)(30,45)(33,46)(34,35)(37,41)(38,43)(40,42);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,33)(21,36)(22,31)(23,29)(24,37)(26,42)(27,39)(28,41)(30,35)(32,45)(34,46)(38,48)(40,44)(43,47);(11,16)(13,15)(17,20,22,32,23,35,21,34)(18,26,27,43,28,44,24,38)(19,30,31,46,29,33,36,45)(25,40,39,48,41,42,37,47);(10,14)(13,15)(17,21,23,22)(18,24,28,27)(19,31,29,36)(20,34,35,32)(25,39,41,37)(26,38,44,43)(30,46,33,45)(40,48,42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6925908);