#8497838

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
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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8405034);