#8922564

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

4336 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

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, 8829760);