#8091740

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	#39444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4412 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)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,21)(22,24)(25,26)(27,31)(28,39)(29,33)(30,34)(32,38)(35,36)(37,40);(1,2,4,5)(3,6,7,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,24,22,23)(25,27)(26,31)(28,37)(29,38)(30,35)(32,33)(34,36)(39,40);(1,3,4,7)(2,6,5,8)(10,12)(13,21)(15,17)(18,22)(27,35)(28,38)(31,36)(32,39);(9,11)(10,15)(12,17)(13,22)(14,19)(16,24)(18,21)(20,23)(25,26)(27,36)(28,39)(29,40)(30,34)(31,35)(32,38)(33,37);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,24,22,23)(25,28,30,38)(26,32,34,39)(27,37,35,29)(31,33,36,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,23)(17,24)(19,22)(25,29,30,37)(26,33,34,40)(27,38,35,28)(31,32,36,39)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,e^2=i,f^2=hi,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, 7998936);