#8214875

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	#56800	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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:

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

Polycyclic presentation:

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

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, 8122071);