#9850813

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

4083 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)(3,7)(4,9)(5,13)(6,8)(10,15)(11,12)(14,16)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(34,40)(36,44)(41,46)(43,45);(1,3,6,9)(2,7,8,4)(5,11,15,16)(10,14,13,12)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,34,39,40)(35,41,37,46)(36,42,44,47)(38,43,48,45);(2,8)(3,9)(10,13)(11,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,35,39,37)(34,41,40,46)(36,43,44,45)(38,42,48,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,16)(13,15)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,36,39,44)(34,42,40,47)(35,43,37,45)(38,41,48,46);(1,4,6,7)(2,9,8,3)(5,14,15,12)(10,11,13,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,39,35)(34,41,40,46)(36,45,44,43)(38,42,48,47);(1,5)(2,10)(3,11)(4,12)(6,15)(7,14)(8,13)(9,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,38,39,48)(34,43,40,45)(35,42,37,47)(36,41,44,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=g,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, 9758009);