#8528139

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

4028 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(18,23)(19,25)(24,30)(26,31)(33,34,38,41)(35,39,44,47)(36,37,40,45)(42,43,46,48);(2,7)(3,9)(10,15)(11,13)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(2,7)(4,8)(10,15)(12,16)(20,27)(24,30)(26,31)(28,32)(33,35,38,44)(34,39,41,47)(36,42,40,46)(37,43,45,48);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36)(34,37)(35,42)(38,40)(39,43)(41,45)(44,46)(47,48);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,16)(9,13)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(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=h,e^2=g,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, 8435335);