#8679960

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	#44935	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4519 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587156);