#8323198

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
prime radical	#59408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4612 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=g,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, 8230394);