#7511096

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1264 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)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,25)(21,34)(22,27)(23,26)(28,38)(29,47)(30,41)(31,40)(32,39)(33,42)(35,45)(36,44)(37,43)(46,48);(1,3)(2,6)(4,7)(5,8)(17,19,23,31)(18,24,26,40)(20,32,21,29)(22,36,37,30)(25,39,34,47)(27,44,43,41)(28,35,46,33)(38,45,48,42);(10,13)(14,15)(17,20,23,21)(18,25,26,34)(19,28,31,46)(22,35,37,33)(24,38,40,48)(27,45,43,42)(29,36,32,30)(39,41,47,44);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,21,23,20)(18,25,26,34)(19,29,31,32)(22,35,37,33)(24,39,40,47)(27,42,43,45)(28,36,46,30)(38,41,48,44);(1,4,5,6)(2,3,7,8)(18,26)(20,21)(24,40)(27,43)(28,46)(29,32)(33,35)(41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=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, 7418292);