#7969687

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	#39416	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4545 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7876883);