#9718263

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

4213 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,24)(19,23,26,20)(21,29,31,25)(27,28,32,30)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,45)(46,47);(3,9)(7,13)(10,15)(14,16)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,29)(31,32)(33,35)(34,37)(36,44)(38,42)(39,41)(40,48)(43,46)(45,47);(2,6)(7,13)(8,11)(14,16)(18,24)(19,26)(25,29)(27,32)(34,39)(35,42)(40,47)(43,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,19)(21,30)(22,23)(24,26)(25,32)(27,29)(28,31)(33,36,38,46)(34,40,39,47)(35,43,42,44)(37,45,41,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,31)(18,25,24,29)(19,27,26,32)(20,28,23,30)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=gi,e^2=ghi,f^2=gh,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, 9625459);