#7791060

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	#45064	C₂
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7698256);