#8093715

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	#52427	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4350 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8000911);