#7609516

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,38)(20,25,30,37)(21,26,35,42)(22,27,33,40)(29,36,45,48)(32,39,44,47)(34,41,43,46);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,28,38)(20,22,30,33)(21,32,35,44)(25,27,37,40)(26,39,42,47)(29,43,45,34)(36,46,48,41);(10,14)(11,15)(17,20,23,30)(18,25,28,37)(19,22,31,33)(21,32,35,44)(24,27,38,40)(26,39,42,47)(29,34,45,43)(36,41,48,46);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,33,43,23,35,22,34)(18,26,40,46,28,42,27,41)(19,29,30,44,31,45,20,32)(24,36,37,47,38,48,25,39);(2,5)(6,8)(17,22,23,33)(18,27,28,40)(19,30,31,20)(21,34,35,43)(24,37,38,25)(26,41,42,46)(29,44,45,32)(36,47,48,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=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, 7516712);