#9723152

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

4092 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)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(33,34,38,42)(35,39,44,48)(36,45,46,40)(37,41,43,47);(1,3,6,11)(2,7,8,14)(4,9,12,15)(5,10,13,16)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(33,35,38,44)(34,39,42,48)(36,37,46,43)(40,47,45,41);(2,8)(4,12)(7,14)(9,15)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,36,38,46)(34,40,42,45)(35,43,44,37)(39,41,48,47);(1,4)(2,5)(3,9)(6,12)(7,10)(8,13)(11,15)(14,16)(17,21,23,20)(18,26,27,25)(19,30,22,29)(24,32,28,31)(33,36)(34,40)(35,37)(38,46)(39,47)(41,48)(42,45)(43,44);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,22,23,19)(18,24,27,28)(20,30,21,29)(25,31,26,32)(33,37)(34,41)(35,36)(38,43)(39,45)(40,48)(42,47)(44,46);(1,5)(2,4)(3,10)(6,13)(7,9)(8,12)(11,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,31)(30,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9630348);