#8437824

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	#54852	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345020);