#8107521

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

4536 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014717);