#8968129

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

4118 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8875325);