#9863853

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

4164 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31)(33,34)(35,39)(36,37)(38,40)(41,45)(42,43)(44,46)(47,48);(2,7)(4,8)(9,15)(12,13)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(33,35,38,44)(34,39,40,46)(36,42,41,47)(37,43,45,48);(18,22)(23,30)(24,27)(31,32)(34,40)(36,41)(39,46)(42,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(33,36,38,41)(34,37,40,45)(35,42,44,47)(39,43,46,48);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,20)(18,24)(19,25)(21,29)(22,27)(23,31)(26,28)(30,32)(33,37)(34,41)(35,43)(36,40)(38,45)(39,47)(42,46)(44,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9771049);