#9851193

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

4127 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,38,41)(35,37,43,39)(36,40,45,48)(42,46,47,44);(3,9)(7,13)(10,15)(14,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,35)(34,37)(36,42)(38,43)(39,41)(40,46)(44,48)(45,47);(2,6)(7,13)(8,11)(14,16)(18,25)(19,29)(20,31)(27,32)(33,35,38,43)(34,39,41,37)(36,42,45,47)(40,44,48,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,46)(44,48)(45,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,36)(34,40)(35,42)(37,46)(38,45)(39,44)(41,48)(43,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,37)(34,35)(36,44)(38,39)(40,47)(41,43)(42,48)(45,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,e^2=ghi,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, 9758389);