#8358291

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	#40728	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4650 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=ghi,e^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,e^d=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, 8265487);