#8455752

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

4360 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gh,e^2=ghi,f^2=gh,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, 8362948);