#8639918

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8547114);