#9939593

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

4288 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9846789);