#8425652

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	#39948	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4422 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332848);