#8471020

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	#37748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4597 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)(3,7)(9,10)(11,18)(12,21)(13,22)(14,16)(15,17)(19,20)(23,24)(25,26,31,28)(27,32,38,35)(29,39,37,33)(30,34,36,40);(1,2,4,5)(3,6,7,8)(9,11,15,14)(10,16,17,18)(12,19,22,24)(13,20,21,23)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,34)(39,40);(1,2)(3,6)(4,5)(7,8)(25,26,31,28)(27,35,38,32)(29,33,37,39)(30,34,36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,19)(14,24)(15,22)(16,20)(17,21)(18,23)(25,28,31,26)(27,35,38,32)(29,39,37,33)(30,34,36,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,12)(11,20)(14,23)(15,21)(16,19)(17,22)(18,24)(25,29,31,37)(26,33,28,39)(27,36,38,30)(32,34,35,40);(9,14,15,11)(10,16,17,18)(12,19,22,24)(13,23,21,20)(25,30,31,36)(26,34,28,40)(27,37,38,29)(32,33,35,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8378216);