#7586603

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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1243 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:

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

Polycyclic presentation:

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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7493799);