#7566555

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1298 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)(9,10,13,16)(11,15,12,14)(17,18,23,28)(19,24,32,39)(20,25,33,40)(21,26,22,27)(29,36,45,48)(30,37,31,38)(34,41,35,42)(43,46,44,47);(2,5)(6,8)(11,12)(14,15)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,36)(26,38)(27,37)(28,39)(33,45)(34,44)(35,43)(40,48)(41,47)(42,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,36)(26,42)(27,41)(28,40)(30,44)(31,43)(32,45)(37,47)(38,46)(39,48);(9,12,13,11)(10,15,16,14)(19,30)(20,33)(21,22)(24,37)(25,40)(26,27)(29,43)(31,32)(36,46)(38,39)(44,45)(47,48);(2,5)(6,8)(17,21,23,22)(18,26,28,27)(19,31,32,30)(20,34,33,35)(24,38,39,37)(25,41,40,42)(29,44,45,43)(36,47,48,46)

Polycyclic presentation:

2^9|a^2=gi,c^2=fg,d^2=g,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,d^c=di,e^c=ei,e^d=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, 7473751);