#7513379

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 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)(17,18,22,27)(19,29,32,24)(20,25,36,43)(21,26,23,28)(30,45,46,38)(31,40,33,39)(34,48,47,41)(35,42,37,44);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19,22,32)(18,24,27,29)(20,34,36,47)(21,33,23,31)(25,41,43,48)(26,40,28,39)(30,37,46,35)(38,44,45,42);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,20,22,36)(18,25,27,43)(19,30,32,46)(21,35,23,37)(24,38,29,45)(26,42,28,44)(31,34,33,47)(39,41,40,48);(2,3)(7,8)(10,13)(14,16)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,35,36,37)(24,39,29,40)(25,42,43,44)(30,34,46,47)(38,41,45,48);(1,5,6,4)(2,8,3,7)(18,27)(24,29)(25,43)(26,28)(38,45)(39,40)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=fgi,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, 7420575);