#8343527

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	#42950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8250723);