#8505412

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

4237 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=gi,e^2=g,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, 8412608);