#8346036

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

4405 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)(10,11)(15,17)(16,18)(22,23)(25,26)(27,31)(28,35)(29,34)(30,40)(32,37)(33,39)(36,38);(2,6)(3,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,33)(28,38)(29,30)(31,37)(32,39)(34,36)(35,40);(1,3)(2,4)(5,8)(6,7)(13,21)(16,18)(19,24)(22,23)(25,28,32,29)(26,34,37,35)(27,38,39,30)(31,40,33,36);(1,3)(2,4)(5,8)(6,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,29,32,28)(26,35,37,34)(27,38,39,30)(31,40,33,36);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,30,32,38)(26,36,37,40)(27,29,39,28)(31,35,33,34);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,31,32,33)(26,27,37,39)(28,36,29,40)(30,35,38,34)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gi,e^2=gi,f^2=gi,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, 8253232);