#9835837

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

4014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,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, 9743033);