#9863935

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
prime radical	#56283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,12)(4,6)(5,8)(7,9)(10,11)(13,16)(14,15)(17,18)(19,23)(20,24)(21,29)(22,26)(25,31)(27,28)(30,32)(34,40)(36,45)(41,47)(43,46);(1,3,7,5)(2,8,9,12)(4,13,11,15)(6,14,10,16)(17,19,22,24)(18,23,26,20)(21,27,31,32)(25,30,29,28)(33,34,39,40)(35,41,37,47)(36,42,45,38)(43,48,46,44);(2,9)(4,11)(8,12)(13,15)(17,18,22,26)(19,20,24,23)(21,25,31,29)(27,28,32,30)(33,35,39,37)(34,41,40,47)(36,43,45,46)(38,44,42,48);(1,4,7,11)(2,10,9,6)(3,13,5,15)(8,16,12,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,36)(34,42)(35,43)(37,46)(38,40)(39,45)(41,48)(44,47);(1,5,7,3)(2,8,9,12)(4,15,11,13)(6,14,10,16)(17,20,22,23)(18,24,26,19)(21,30,31,28)(25,27,29,32)(33,37,39,35)(34,41,40,47)(36,46,45,43)(38,44,42,48);(1,6,7,10)(2,11,9,4)(3,14,5,16)(8,15,12,13)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,32)(26,29)(33,38,39,42)(34,36,40,45)(35,44,37,48)(41,43,47,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,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, 9771131);