#9812900

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

4235 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=gi,e^2=g,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, 9720096);