#8471639

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	#40510	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,19)(13,16)(14,15)(18,24)(20,21)(22,23)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(2,5)(6,7)(10,15)(12,17)(16,23)(20,24)(25,27,31,28)(26,32,33,36)(29,37,39,34)(30,35,40,38);(1,2)(3,6)(4,5)(7,8)(25,28,31,27)(26,32,33,36)(29,37,39,34)(30,38,40,35);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,21)(14,19)(15,17)(16,24)(18,22)(20,23)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378835);