#8930909

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

4130 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8838105);