#8806898

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
prime radical	#45655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714094);