#7621161

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1282 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)(19,29)(20,33)(21,26)(22,27)(23,28)(24,31)(25,36)(30,39)(32,43)(34,40)(35,46)(37,47)(38,44)(41,45)(42,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,29)(20,34,36,30)(21,38,32,23)(25,40,33,39)(26,28,43,44)(35,37,45,48)(41,47,46,42);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,39)(26,41)(27,33)(28,42)(29,40)(31,34)(32,45)(38,48)(43,46)(44,47);(10,13)(14,16)(17,21,23,19,22,32,38,31)(18,26,28,29,27,43,44,24)(20,35,37,30,36,45,48,34)(25,41,42,40,33,46,47,39);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(18,27)(20,36)(24,29)(26,43)(28,44)(30,34)(35,45)(37,48)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=ghi,e^2=fg,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, 7528357);