#7566402

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1300 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)(17,18,22,25)(19,23,30,36)(20,24,31,39)(21,26,33,40)(27,34,42,47)(28,35,29,37)(32,38,44,48)(41,45,43,46);(2,5)(6,8)(11,12)(14,15)(17,19,22,30)(18,23,25,36)(20,27,31,42)(21,28,33,29)(24,34,39,47)(26,35,40,37)(32,41,44,43)(38,45,48,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,39)(26,38)(28,41)(29,43)(30,42)(33,44)(35,45)(36,47)(37,46)(40,48);(9,12,13,11)(10,15,16,14)(19,28)(21,33)(23,35)(26,40)(27,41)(29,30)(32,44)(34,45)(36,37)(38,48)(42,43)(46,47);(2,5)(6,8)(18,25)(20,31)(23,36)(26,40)(27,42)(32,44)(35,37)(41,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fg,d^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, 7473598);