#7621048

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

1287 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)(17,18,22,27)(19,29,31,24)(20,25,34,40)(21,26,32,43)(23,28,36,44)(30,45,46,38)(33,39,37,42)(35,41,47,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,29)(20,30,34,46)(21,36,32,23)(25,38,40,45)(26,28,43,44)(33,35,37,47)(39,48,42,41);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,30)(21,37)(22,34)(23,35)(24,38)(26,42)(27,40)(28,41)(29,45)(31,46)(32,33)(36,47)(39,43)(44,48);(10,13)(14,16)(17,21,23,19,22,32,36,31)(18,26,28,29,27,43,44,24)(20,33,35,46,34,37,47,30)(25,39,41,38,40,42,48,45);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(18,27)(20,34)(24,29)(26,43)(28,44)(30,46)(33,37)(35,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,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,d^c=di,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, 7528244);