#7865064

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

4197 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26)(27,31)(28,29)(30,38)(32,34)(33,39)(35,36)(37,40);(1,3,5,8)(2,6,4,7)(25,27,32,29)(26,31,34,28)(30,35,39,37)(33,40,38,36);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,22)(18,23)(25,28,32,31)(26,27,34,29)(30,36,39,40)(33,35,38,37);(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,29,32,27)(26,28,34,31)(30,37,39,35)(33,36,38,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,40)(32,39)(34,38);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,31,32,28)(26,27,34,29)(30,36,39,40)(33,37,38,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=h,f^2=h,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, 7772260);