#9267003

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

4007 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,23,24)(19,25,20,29)(21,22,26,32)(27,31,30,28);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(18,24)(22,32)(25,29)(28,31)(33,34,39,41)(35,42,36,37)(38,46,47,40)(43,45,44,48);(2,7)(3,10)(8,15)(11,12)(17,19,23,20)(18,25,24,29)(21,30,26,27)(22,28,32,31)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,46,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,23,19)(18,25,24,29)(21,30,26,27)(22,31,32,28)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,46,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,21,23,26)(18,22,24,32)(19,27,20,30)(25,31,29,28)(33,37)(34,35)(36,41)(38,45)(39,42)(40,43)(44,46)(47,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22)(18,26)(19,28)(20,31)(21,24)(23,32)(25,27)(29,30)(33,38)(34,40)(35,43)(36,44)(37,45)(39,47)(41,46)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174199);