#8751575

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	#2819	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4141 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8658771);