#7870496

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4553 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=i,e^2=gh,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, 7777692);