#8492796

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

4089 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,23,24)(19,25,20,30)(21,22,27,26)(28,29,31,32)(33,34,39,37)(35,42,40,36)(38,41,47,45)(43,46,48,44);(2,3)(7,9)(8,10)(14,15)(18,24)(21,27)(25,30)(28,31)(34,37)(36,42)(41,45)(44,46);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,23,20)(18,25,24,30)(21,28,27,31)(22,29,26,32)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,25,24,30)(21,31,27,28)(22,29,26,32)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,27)(18,26,24,22)(19,28,20,31)(25,32,30,29)(33,37,39,34)(35,42,40,36)(38,45,47,41)(43,46,48,44);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,26)(18,27,24,21)(19,29,20,32)(25,31,30,28)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=g,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, 8399992);