#8425596

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

4550 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,e^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

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, 8332792);