#8087824

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7995020);