#8087939

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	#47210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=h,e^2=h,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, 7995135);