#7614843

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,47)(32,42)(33,40)(34,43)(35,41)(36,45)(37,39)(38,46)(44,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,29)(20,30,33,37)(21,34,36,48)(23,35,38,32)(25,39,40,47)(26,44,45,43)(28,42,46,41);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,33)(18,25,27,40)(19,30,31,37)(21,35,36,32)(23,34,38,48)(24,39,29,47)(26,41,45,42)(28,43,46,44);(10,14)(13,15)(17,21,23,37,22,36,38,30)(18,26,28,39,27,45,46,47)(19,20,32,34,31,33,35,48)(24,40,41,44,29,25,42,43);(2,5)(6,8)(10,14)(13,15)(18,27)(20,33)(21,36)(24,29)(28,46)(30,37)(34,48)(41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=h,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

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, 7522039);