#7987061

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(9,10,13,14)(11,17,18,15)(12,16,22,20)(19,24,21,23)(25,26,30,29)(27,31,35,33)(28,36,37,32)(34,39,40,38);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,18)(19,24)(21,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(10,14)(11,18)(16,20)(19,21)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,4,5,3)(2,8,6,7)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=gi,e^2=g,f^2=i,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, 7894257);