#7512342

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1550 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,34)(22,26)(23,27)(28,37)(29,40)(30,39)(31,41)(32,44)(33,43)(35,38)(36,42)(45,48)(46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,27,39)(20,31,33,45)(21,29,22,35)(25,41,43,48)(26,38,34,40)(28,36,46,32)(37,42,47,44);(10,14)(13,15)(17,20,23,33)(18,25,27,43)(19,28,30,46)(21,32,22,36)(24,37,39,47)(26,42,34,44)(29,45,35,31)(38,41,40,48);(2,5)(6,8)(11,16)(13,15)(17,21,23,22)(18,26,27,34)(19,29,30,35)(20,32,33,36)(24,38,39,40)(25,42,43,44)(28,45,46,31)(37,41,47,48);(1,3,4,7)(2,6,5,8)(18,27)(24,39)(25,43)(26,34)(37,47)(38,40)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi

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, 7419538);