#8493226

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

4261 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)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34,39,42)(35,43,46,40)(36,47,48,37)(38,41,45,44);(2,4)(7,9)(8,12)(15,16)(19,20)(21,29)(23,26)(24,31)(33,35)(34,40)(36,44)(37,38)(39,46)(41,48)(42,43)(45,47);(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,36)(34,37)(35,44)(38,40)(39,48)(41,46)(42,47)(43,45);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,36)(34,37)(35,41)(38,43)(39,48)(40,45)(42,47)(44,46);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,36)(35,45)(38,46)(39,47)(40,41)(42,48)(43,44);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,38,39,45)(34,41,42,44)(35,37,46,47)(36,40,48,43)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=gh,e^2=ghi,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, 8400422);