#8091723

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	#46352	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4493 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=ghi,f^2=gi,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, 7998919);