#7544199

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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)(19,23)(20,24)(21,26)(22,25)(27,35)(28,37)(29,36)(30,38)(31,39)(32,40)(33,41)(34,42)(43,46)(44,47)(45,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(20,30)(21,33)(24,38)(26,41)(27,43)(29,44)(31,45)(32,34)(35,46)(36,47)(39,48)(40,42);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,22,28)(18,23,25,37)(20,31,32,27)(21,29,33,44)(24,39,40,35)(26,36,41,47)(30,45,34,43)(38,48,42,46);(10,14)(13,15)(17,20,22,32)(18,24,25,40)(19,27,28,31)(21,34,33,30)(23,35,37,39)(26,42,41,38)(29,45,44,43)(36,48,47,46);(2,5)(6,8)(10,14)(13,15)(18,25)(19,28)(24,40)(26,41)(27,31)(29,44)(38,42)(43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fgi,d^2=i,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,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, 7451395);