#8461196

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

4609 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:

(1,2,5,8)(3,4,7,6)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,2)(3,6)(4,7)(5,8)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,2)(3,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,4)(2,7)(3,8)(5,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(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=ghi,e^2=ghi,f^2=i,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, 8368392);