#8471692

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

4114 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8378888);