#8464082

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

4082 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,e^2=gi,f^2=i,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, 8371278);