#8551306

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

4110 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8458502);