#7555368

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,34)(21,26)(22,27)(23,28)(24,29)(30,43)(31,40)(32,41)(33,42)(35,39)(36,45)(37,44)(38,46)(47,48);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,35)(21,30)(22,32)(23,33)(24,38)(26,43)(27,41)(28,42)(29,46)(31,37)(34,39)(36,47)(40,44)(45,48);(10,14)(11,16)(17,20,23,21)(18,26,28,34)(19,30,33,35)(22,37,24,36)(25,39,42,43)(27,45,29,44)(31,32,47,38)(40,46,48,41);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,20)(18,26,28,34)(19,31,33,47)(22,37,24,36)(25,40,42,48)(27,44,29,45)(30,32,35,38)(39,46,43,41);(2,7)(6,8)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,32,33,38)(20,36,21,37)(25,41,42,46)(26,44,34,45)(30,31,35,47)(39,48,43,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,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, 7462564);