#9782608

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

4129 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)(18,19)(23,25)(24,26)(31,32)(33,34,38,39)(35,37,41,46)(36,40,44,47)(42,45,48,43);(3,9)(7,13)(10,15)(14,16)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(33,35)(34,37)(36,42)(38,41)(39,46)(40,45)(43,47)(44,48);(2,6)(7,13)(8,11)(14,16)(34,39)(35,41)(40,47)(42,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,35)(34,37)(36,42)(38,41)(39,46)(40,45)(43,47)(44,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,37)(34,35)(36,43)(38,46)(39,41)(40,48)(42,47)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=g,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, 9689804);