#8120058

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

4472 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8027254);