#9851014

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

3941 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,23,20)(19,21,24,26)(22,28,31,25)(27,30,32,29)(33,34)(35,45)(36,39)(37,46)(38,42)(40,44)(41,48)(43,47);(2,7)(4,8)(9,15)(12,13)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,35)(34,41)(36,46)(37,42)(38,43)(39,47)(40,48)(44,45);(18,20)(19,24)(25,28)(27,32)(33,36,40,38)(34,42,44,39)(35,46,48,43)(37,45,47,41);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,37,40,47)(34,43,44,46)(35,42,48,39)(36,41,38,45);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,38,40,36)(34,42,44,39)(35,43,48,46)(37,45,47,41);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,39,40,42)(34,38,44,36)(35,47,48,37)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=h,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, 9758210);