#9685515

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

4088 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,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, 9592711);