#10368013

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

4653 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275209);