#8437810

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8345006);