#9923644

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830840);