#8093199

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	#45976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4460 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000395);