#7868927

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4272 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7776123);