#8528131

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

4016 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)(3,9)(4,10)(5,8)(6,7)(11,16)(12,13)(14,15)(18,19)(23,25)(24,26)(30,31)(33,34,38,41)(35,39,44,47)(36,37,40,45)(42,43,46,48);(2,7)(3,10)(8,15)(11,13)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(2,7)(4,9)(8,15)(12,16)(20,27)(23,25)(28,32)(30,31)(33,35,38,44)(34,39,41,47)(36,42,40,46)(37,43,45,48);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36)(34,37)(35,42)(38,40)(39,43)(41,45)(44,46)(47,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,16)(10,13)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,43)(36,41)(38,45)(39,46)(42,47)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8435327);