#8456033

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

4578 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)(11,19)(12,23)(13,24)(14,18)(15,22)(16,21)(17,20)(25,26,32,35)(27,33,30,31)(28,38,29,34)(36,39,37,40);(3,7)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(25,27,32,30)(26,33,35,31)(28,37,29,36)(34,39,38,40);(1,2)(3,6)(4,5)(7,8)(25,28,32,29)(26,34,35,38)(27,36,30,37)(31,39,33,40);(1,3,4,7)(2,6,5,8)(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,17,19,24)(25,29,32,28)(26,34,35,38)(27,37,30,36)(31,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,16,19,23)(25,30,32,27)(26,31,35,33)(28,37,29,36)(34,39,38,40);(9,10)(11,15)(12,23)(13,17)(14,18)(16,21)(19,22)(20,24)(25,31)(26,27)(28,39)(29,40)(30,35)(32,33)(34,36)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gh,e^2=gi,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, 8363229);