#8472498

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

4575 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=h,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, 8379694);