#7896100

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	#2819	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=i,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7803296);