#7616329

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(3,7)(4,5)(6,8)(11,12)(14,15)(17,18,22,26)(19,23,28,34)(20,24,31,37)(21,25,32,38)(27,33,40,44)(29,35,41,45)(30,36,42,46)(39,43,47,48);(2,5)(6,7)(9,10,13,16)(11,15,12,14)(20,30)(21,32)(24,36)(25,38)(27,39)(29,41)(31,42)(33,43)(35,45)(37,46)(40,47)(44,48);(9,11,13,12)(10,14,16,15)(17,19,22,28)(18,23,26,34)(20,27,31,40)(21,29,32,41)(24,33,37,44)(25,35,38,45)(30,39,42,47)(36,43,46,48);(9,12,13,11)(10,15,16,14)(17,20,21,30,22,31,32,42)(18,24,25,36,26,37,38,46)(19,27,29,39,28,40,41,47)(23,33,35,43,34,44,45,48);(1,3,4,8)(2,6,5,7)(19,28)(20,31)(23,34)(24,37)(29,41)(30,42)(35,45)(36,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,e^2=f,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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7523525);