#8460711

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
prime radical	#53653	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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)(3,6)(4,7)(5,8)(9,10)(11,20)(12,17)(13,23)(14,24)(15,19)(16,22)(18,21)(25,26,32,30)(27,37,39,33)(28,35,29,34)(31,36,40,38);(3,8)(5,6)(9,11,15,22)(10,16,19,20)(12,21,13,14)(17,24,23,18)(25,27,32,39)(26,33,30,37)(28,38,29,36)(31,35,40,34);(1,3)(2,5)(4,8)(6,7)(9,12,15,13)(10,17,19,23)(11,21,22,14)(16,24,20,18)(25,28,32,29)(26,34,30,35)(27,38,39,36)(31,37,40,33);(1,3)(2,6)(4,8)(5,7)(9,13,15,12)(10,17,19,23)(11,14,22,21)(16,24,20,18)(25,29,32,28)(26,35,30,34)(27,38,39,36)(31,37,40,33);(9,14)(10,18)(11,12)(13,22)(15,21)(16,17)(19,24)(20,23)(25,30,32,26)(27,37,39,33)(28,35,29,34)(31,38,40,36);(1,2)(3,5)(4,7)(6,8)(9,10)(11,16)(12,17)(13,23)(14,18)(15,19)(20,22)(21,24)(25,31,32,40)(26,36,30,38)(27,35,39,34)(28,37,29,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=hi,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, 8367907);