#8197027

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	#34976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(2,5)(6,7)(9,10,14,15)(11,13,17,22)(12,16,21,19)(18,20,23,24)(26,31)(27,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,22)(16,19)(20,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104223);