#7620082

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 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)(4,6)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18,21,25)(19,27,28,23)(20,24,33,38)(22,26,32,37)(29,39,41,34)(30,40,42,35)(31,36,45,48)(43,47,44,46);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(19,28)(20,31)(22,32)(23,27)(24,36)(26,37)(29,43)(33,45)(34,46)(38,48)(39,47)(41,44);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,19,21,28)(18,23,25,27)(20,29,33,41)(22,30,32,42)(24,34,38,39)(26,35,37,40)(31,44,45,43)(36,47,48,46);(9,11,13,12)(10,15,14,16)(17,20,32,45,21,33,22,31)(18,24,37,48,25,38,26,36)(19,29,42,43,28,41,30,44)(23,34,40,46,27,39,35,47);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(19,28)(23,27)(29,41)(30,42)(34,39)(35,40)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=hi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527278);