#7710977

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 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)(9,10,15,14)(11,18,13,16)(12,17,22,21)(19,24,20,23)(28,36)(32,37)(34,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,14,18)(12,19,22,20)(17,23,21,24)(26,30)(31,33)(32,37)(38,39);(9,11,15,13)(10,16,14,18)(12,20,22,19)(17,24,21,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24);(1,2,5,4)(3,6,8,7)(9,13,15,11)(10,18,14,16)(12,20,22,19)(17,24,21,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,5,2)(3,7,8,6)(9,14,15,10)(11,18,13,16)(12,21,22,17)(19,24,20,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618173);