#7510824

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

1266 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)(19,23)(20,30)(21,25)(22,26)(24,32)(27,45)(28,38)(29,39)(31,48)(33,44)(34,42)(35,43)(36,41)(37,46)(40,47);(1,3)(2,6)(4,7)(5,8)(17,19,21,28)(18,23,25,38)(20,31,32,47)(22,35,34,29)(24,40,30,48)(26,43,42,39)(27,36,46,33)(37,44,45,41);(10,13)(14,15)(17,20,21,32)(18,24,25,30)(19,27,28,46)(22,36,34,33)(23,37,38,45)(26,44,42,41)(29,47,35,31)(39,48,43,40);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(18,25)(23,38)(24,30)(26,42)(37,45)(39,43)(40,48)(41,44);(1,4,5,6)(2,3,7,8)(18,25)(20,32)(23,38)(26,42)(27,46)(31,47)(33,36)(39,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7418020);