#8475714

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

4477 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382910);