#8456001

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

4219 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18,22,26)(19,24,27,23)(20,21,25,31)(28,30,32,29)(34,40)(35,36)(42,46)(43,45);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(19,27)(23,24)(28,32)(29,30)(33,34,39,40)(35,41,36,44)(37,42,47,46)(38,43,48,45);(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,32)(30,31)(33,35,39,36)(34,41,40,44)(37,43,47,45)(38,42,48,46);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,19)(18,24)(20,28)(21,30)(22,27)(23,26)(25,32)(29,31)(33,36,39,35)(34,41,40,44)(37,45,47,43)(38,42,48,46);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,20,22,25)(18,21,26,31)(19,28,27,32)(23,29,24,30)(33,37,39,47)(34,42,40,46)(35,43,36,45)(38,41,48,44);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,21)(18,25)(19,29)(20,26)(22,31)(23,32)(24,28)(27,30)(33,38,39,48)(34,43,40,45)(35,42,36,46)(37,41,47,44)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,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,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, 8363197);