#8463527

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

4009 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370723);