#7566311

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

1305 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)(17,18,22,27)(19,28,32,24)(20,25,34,41)(21,35,23,26)(29,43,45,37)(30,39,31,38)(33,42,36,40)(44,48,46,47);(2,5)(6,8)(11,12)(14,15)(17,19,22,32)(18,24,27,28)(20,29,34,45)(21,31,23,30)(25,37,41,43)(26,38,35,39)(33,44,36,46)(40,48,42,47);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,29)(21,36)(22,34)(23,33)(24,37)(26,42)(27,41)(28,43)(30,44)(31,46)(32,45)(35,40)(38,47)(39,48);(9,12,13,11)(10,15,16,14)(19,30)(21,23)(24,38)(26,35)(28,39)(29,44)(31,32)(33,36)(37,47)(40,42)(43,48)(45,46);(2,5)(6,8)(17,21,22,23)(18,26,27,35)(19,31,32,30)(20,33,34,36)(24,38,28,39)(25,40,41,42)(29,44,45,46)(37,48,43,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,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,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, 7473507);