#8757398

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

4187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664594);