#8188108

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	#52427	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 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)(4,5)(6,8)(9,10,14,15)(11,17,19,13)(12,20,22,16)(18,21,24,23)(25,26,31,34)(27,28,29,32)(30,33,38,40)(35,36,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,23)(22,24)(25,27,31,29)(26,32,34,28)(30,35,38,37)(33,39,40,36);(2,5)(6,7)(10,15)(13,17)(16,20)(21,23)(30,38)(33,40)(35,37)(36,39);(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,23)(22,24)(25,28,31,32)(26,27,34,29)(30,36,38,39)(33,35,40,37);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,16,15,20)(11,18,19,24)(13,21,17,23)(25,29,31,27)(26,32,34,28)(30,37,38,35)(33,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,21)(14,17)(15,19)(16,18)(20,24)(22,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095304);