#8712454

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

4177 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)(3,7)(4,13)(5,10)(6,8)(9,14)(11,15)(12,16)(17,18,23,25)(19,27,21,24)(20,22,28,26)(29,30,31,32)(33,34)(35,42)(36,44)(37,39)(38,41)(40,45)(43,47)(46,48);(1,3,6,11)(2,7,8,15)(4,12,14,10)(5,13,16,9)(17,19,23,21)(18,24,25,27)(20,31,28,29)(22,30,26,32)(33,35,38,37)(34,39,41,42)(36,43,45,46)(40,47,44,48);(2,8)(3,11)(9,13)(10,12)(18,25)(20,28)(24,27)(29,31)(36,45)(40,44)(43,46)(47,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,15)(10,14)(13,16)(17,20)(18,26)(19,29)(21,31)(22,25)(23,28)(24,30)(27,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,44)(42,48);(1,4,6,14)(2,9,8,13)(3,12,11,10)(5,15,16,7)(17,21,23,19)(18,27,25,24)(20,31,28,29)(22,30,26,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,45)(46,48);(1,5)(2,10)(3,13)(4,7)(6,16)(8,12)(9,11)(14,15)(17,22,23,26)(18,28,25,20)(19,30,21,32)(24,29,27,31)(33,37,38,35)(34,39,41,42)(36,43,45,46)(40,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619650);