#8720812

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

4226 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32)(33,34,39,44)(35,45,47,40)(36,38,43,41)(37,48,46,42);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,18)(19,26)(20,24)(21,25)(22,23)(27,30)(28,29)(31,32)(33,35)(34,40)(36,37)(38,42)(39,47)(41,48)(43,46)(44,45);(4,13)(8,12)(11,14)(15,16)(18,23)(19,24)(25,32)(27,29)(33,36,39,43)(34,41,44,38)(35,37,47,46)(40,48,45,42);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,19,22,24)(18,20,23,26)(21,29,31,27)(25,30,32,28)(33,36)(34,41)(35,46)(37,47)(38,44)(39,43)(40,42)(45,48);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,37,39,46)(34,42,44,48)(35,36,47,43)(38,45,41,40);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,21,22,31)(18,25,23,32)(19,27,24,29)(20,28,26,30)(33,38,39,41)(34,43,44,36)(35,42,47,48)(37,40,46,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=i,e^2=g,f^2=h,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, 8628008);