#8116771

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	#34987	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4500 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8023967);