#8898647

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

4161 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34)(35,40)(36,46)(37,41)(38,44)(39,43)(42,48)(45,47);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,39,45)(34,40,43,47)(36,44,37,42)(38,41,48,46);(2,6)(8,9)(10,13)(15,16)(21,29)(24,31)(26,32)(28,30)(33,36,39,37)(34,41,43,46)(35,42,45,44)(38,47,48,40);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,34)(35,40)(36,41)(37,46)(38,42)(39,43)(44,48)(45,47);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37,39,36)(34,41,43,46)(35,44,45,42)(38,47,48,40);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38)(34,42)(35,41)(36,40)(37,47)(39,48)(43,44)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8805843);