#8094735

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

4403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 8001931);