#9727143

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

4092 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9634339);