#8434762

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

4616 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341958);