#8703077

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	#47633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4397 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8610273);