#8994391

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

4132 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(17,18,23,21)(19,20,27,24)(22,29,30,25)(26,31,32,28)(33,34,38,42)(35,37,41,39)(36,40,46,44)(43,45,48,47);(1,3,7,4)(2,8,9,10)(5,14,15,12)(6,13,16,11)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(2,9)(3,4)(5,15)(6,16)(18,21)(20,24)(22,30)(26,32)(33,34,38,42)(35,39,41,37)(36,44,46,40)(43,45,48,47);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,45)(47,48);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,21,23,18)(19,24,27,20)(22,29,30,25)(26,31,32,28)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,22,23,30)(18,25,21,29)(19,26,27,32)(20,28,24,31)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=i,e^2=h,f^2=g,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, 8901587);