#8345975

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

4442 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8253171);