#9863018

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

4252 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=i,e^2=ghi,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, 9770214);