#8093858

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	#45578	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4501 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:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,39,38,37);(1,3,4,8)(2,5,7,6)(11,17)(15,21)(18,23)(22,24)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,37,40,39);(1,3)(2,6)(4,8)(5,7)(10,14)(15,21)(16,19)(22,24)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,37,40,39);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(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=gi,d^2=g,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, 8001054);