#7584334

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fg,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7491530);