#8193912

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

4563 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8101108);