#7620948

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

1283 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)(11,14)(12,13)(15,16)(17,18,24,30)(19,25,33,40)(20,26,36,43)(21,27,32,39)(22,28,23,29)(31,38,46,48)(34,41,45,47)(35,42,37,44);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,33)(18,25,30,40)(20,31,36,46)(21,22,32,23)(26,38,43,48)(27,28,39,29)(34,37,45,35)(41,44,47,42);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,34)(22,37)(23,35)(24,36)(25,38)(27,41)(28,44)(29,42)(30,43)(32,45)(33,46)(39,47)(40,48);(10,13)(14,16)(17,21,23,19,24,32,22,33)(18,27,29,25,30,39,28,40)(20,34,35,31,36,45,37,46)(26,41,42,38,43,47,44,48);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,30,29)(19,32,33,21)(20,35,36,37)(25,39,40,27)(26,42,43,44)(31,34,46,45)(38,41,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7528144);