#8463624

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

4022 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,e^2=hi,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, 8370820);