#8657957

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

4502 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,f^2=g,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, 8565153);