#8471647

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	#37789	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(2,5)(6,7)(10,15)(12,22)(16,19)(18,24)(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);(1,3,4,8)(2,6,5,7)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,23)(20,24)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36);(9,13)(10,18)(11,17)(12,16)(14,23)(15,24)(19,22)(20,21)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,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, 8378843);