#9850642

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

4114 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,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, 9757838);