#7987168

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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)(11,14)(12,20)(13,15)(16,21)(17,19)(18,24)(22,23)(25,26,30,29)(27,31,35,33)(28,36,37,32)(34,39,40,38);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,24)(21,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(10,13)(11,17)(12,21)(22,24)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3,5,4)(2,7,6,8)(9,12,15,21)(10,16,13,20)(11,18,17,23)(14,22,19,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,4,5,3)(2,8,6,7)(9,13,15,10)(11,19,17,14)(12,20,21,16)(18,24,23,22)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,14,15,19)(10,11,13,17)(12,22,21,24)(16,18,20,23)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=gi,e^2=gh,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, 7894364);