#8898577

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

4145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=h,e^2=g,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, 8805773);