#8095329

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4285 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)(13,21)(16,20)(18,24)(22,23)(25,26)(27,32)(28,35)(29,34)(30,33)(31,36)(37,39)(38,40);(1,3,4,8)(2,5,7,6)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,27)(26,29)(28,37)(30,38)(31,34)(32,36)(33,39)(35,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,32)(28,37)(29,36)(30,38)(31,34)(33,40)(35,39);(1,2)(3,6)(4,7)(5,8)(25,28,31,30)(26,33,36,35)(27,37,34,38)(29,39,32,40);(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,29,31,32)(26,34,36,27)(28,39,30,40)(33,38,35,37);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,28)(26,35,36,33)(27,38,34,37)(29,40,32,39)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=g,e^2=ghi,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, 8002525);