#8492788

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

3941 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,e^2=gi,f^2=g,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, 8399984);