#7513382

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	#35212	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1554 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)(3,6)(4,7)(5,8)(10,13)(11,15)(17,18,23,27)(19,24,30,39)(20,25,33,43)(21,34,22,26)(28,37,46,47)(29,40,35,38)(31,41,45,48)(32,44,36,42);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,31)(21,29)(22,35)(23,30)(25,41)(26,38)(27,39)(28,32)(33,45)(34,40)(36,46)(37,44)(42,47)(43,48);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(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);(2,3)(7,8)(10,13)(14,16)(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);(1,5,6,4)(2,8,3,7)(18,27)(24,39)(25,43)(26,34)(37,47)(38,40)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,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, 7420578);