#8997125

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

4245 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8904321);