#9851056

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

3942 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(33,34,38,42)(35,39,43,47)(36,40,46,45)(37,41,48,44);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,16)(13,15)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(35,43)(36,46)(39,47)(40,45);(2,7)(3,9)(10,13)(11,16)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,35,38,43)(34,39,42,47)(36,44,46,41)(37,45,48,40);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,36,38,46)(34,40,42,45)(35,44,43,41)(37,47,48,39);(1,5)(2,10)(3,11)(4,12)(6,15)(7,13)(8,14)(9,16)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,37)(34,41)(35,45)(36,39)(38,48)(40,43)(42,44)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gh,e^2=i,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, 9758252);