#8459102

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

4656 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8366298);