#8992909

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

4216 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=ghi,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, 8900105);