#9794295

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

3945 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,24,21)(19,22,25,20)(23,29,31,26)(27,28,32,30)(34,40)(35,44)(36,42)(38,47);(2,7)(4,8)(9,15)(12,13)(17,19)(18,22)(20,21)(23,27)(24,25)(26,30)(28,29)(31,32)(33,34,39,40)(35,41,44,48)(36,37,42,45)(38,43,47,46);(17,20,24,22)(18,25,21,19)(23,28,31,30)(26,32,29,27)(33,35,39,44)(34,41,40,48)(36,46,42,43)(37,38,45,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,21,24,18)(19,20,25,22)(23,29,31,26)(27,28,32,30)(33,36,39,42)(34,37,40,45)(35,43,44,46)(38,41,47,48);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,22,24,20)(18,25,21,19)(23,28,31,30)(26,27,29,32)(33,37)(34,42)(35,38)(36,40)(39,45)(41,43)(44,47)(46,48);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,23,24,31)(18,26,21,29)(19,27,25,32)(20,28,22,30)(33,38)(34,43)(35,45)(36,41)(37,44)(39,47)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9701491);