#8471285

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	#38020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4776 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^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, 8378481);