#7536468

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

1579 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)(9,10,12,14)(11,15,16,13)(17,18)(19,28)(20,25)(21,35)(22,27)(23,26)(24,32)(29,43)(30,39)(31,38)(33,42)(34,41)(36,40)(37,45)(44,48)(46,47);(2,5)(6,8)(11,16)(13,15)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,37)(26,38)(27,28)(33,44)(34,45)(35,39)(36,46)(40,48)(41,43)(42,47);(9,11,12,16)(10,13,14,15)(17,20,22,34)(18,25,27,41)(19,29,32,45)(21,36,23,33)(24,37,28,43)(26,42,35,40)(30,44,31,46)(38,47,39,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(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=g,c^2=gi,d^2=fi,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, 7443664);