#9815016

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

4088 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=hi,e^2=g,f^2=gh,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, 9722212);