#8581782

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	#39458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4179 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)(3,5)(4,7)(6,8)(9,10,15,12)(11,13,16,18)(14,17,22,20)(19,21,23,24)(27,35)(28,37)(31,38)(32,40);(2,3)(6,7)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,24)(22,23)(25,26,30,34)(27,31,35,38)(28,32,37,40)(29,33,36,39);(1,2,5,3)(4,6,8,7)(9,10,15,12)(11,18,16,13)(14,17,22,20)(19,24,23,21)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38);(1,3,5,2)(4,7,8,6)(9,12,15,10)(11,18,16,13)(14,17,22,20)(19,21,23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,8)(2,6,3,7)(9,13,15,18)(10,16,12,11)(14,21,22,24)(17,23,20,19)(25,28,30,37)(26,32,34,40)(27,36,35,29)(31,39,38,33);(9,14,15,22)(10,17,12,20)(11,19,16,23)(13,21,18,24)(25,29,30,36)(26,33,34,39)(27,37,35,28)(31,40,38,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=i,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8488978);