#7577168

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,27)(19,29,32,24)(20,25,34,40)(21,26,23,28)(30,43,44,37)(31,42,36,38)(33,39,35,41)(45,48,46,47);(2,6)(7,8)(10,13)(11,16)(17,19,22,32)(18,24,27,29)(20,30,34,44)(21,31,23,36)(25,37,40,43)(26,38,28,42)(33,45,35,46)(39,47,41,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,30)(21,35)(22,34)(23,33)(24,37)(26,41)(27,40)(28,39)(29,43)(31,46)(32,44)(36,45)(38,48)(42,47);(10,13)(14,15)(17,21)(18,26)(20,33)(22,23)(25,39)(27,28)(30,44)(31,36)(34,35)(37,43)(38,42)(40,41);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,33,34,35)(24,38,29,42)(25,39,40,41)(30,45,44,46)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7484364);