#8122402

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4417 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)(6,8)(9,10)(11,14)(12,20)(13,17)(15,19)(16,21)(18,23)(22,24)(25,26,32,28)(27,29,34,31)(30,33,39,36)(35,37,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27)(26,31)(28,29)(30,35)(32,34)(33,38)(36,37)(39,40);(2,5)(6,8)(9,11,13,19)(10,15,17,14)(12,18,21,24)(16,23,20,22)(25,28,32,26)(27,29,34,31)(30,33,39,36)(35,38,40,37);(9,10)(11,15)(12,16)(13,17)(14,19)(18,23)(20,21)(22,24)(25,29,32,31)(26,27,28,34)(30,37,39,38)(33,35,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,30,32,39)(26,33,28,36)(27,35,34,40)(29,37,31,38);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,16,17,20)(11,18,19,24)(14,22,15,23)(25,31,32,29)(26,34,28,27)(30,38,39,37)(33,40,36,35)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8029598);