#8968036

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

4646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=hi,e^2=g,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, 8875232);