#8460674

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

4386 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)(3,4)(5,6)(7,8)(9,10)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(25,26,30,34)(27,29,33,31)(28,32,37,40)(35,36,39,38);(2,6)(3,7)(10,15)(11,18)(16,22)(19,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3,5,7)(2,4,6,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,26)(27,29)(28,32)(30,34)(31,33)(35,36)(37,40)(38,39);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28,30,37)(26,32,34,40)(27,35,33,39)(29,36,31,38);(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,29)(26,33)(27,34)(28,36)(30,31)(32,39)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=gi,f^2=ghi,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, 8367870);