#8602875

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510071);