#7864991

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	#43979	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26)(27,35)(28,33)(29,36)(30,34)(31,32)(37,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27,31,36)(26,29,32,35)(28,30,37,40)(33,38,39,34);(1,3,5,4)(2,7,6,8)(10,16)(15,18)(17,20)(23,24)(26,32)(27,36)(28,37)(34,38);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,28,31,37)(26,33,32,39)(27,30,36,40)(29,38,35,34);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,29,31,35)(26,27,32,36)(28,38,37,34)(30,39,40,33);(1,4,5,3)(2,8,6,7)(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,30)(26,34)(27,37)(28,36)(29,33)(31,40)(32,38)(35,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=gh,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, 7772187);