#8696697

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

4589 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603893);