#7865104

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

4256 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)(25,26,31,36)(27,29,38,32)(28,34,39,33)(30,35,37,40);(1,3,5,8)(2,6,4,7)(25,27)(26,32)(28,30)(29,36)(31,38)(33,35)(34,40)(37,39);(10,15)(11,16)(17,22)(18,23)(25,28,31,39)(26,33,36,34)(27,37,38,30)(29,35,32,40);(1,2,5,4)(3,6,8,7)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,28)(26,34)(27,30)(29,35)(31,39)(32,40)(33,36)(37,38);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,31,32)(26,27,36,38)(28,35,39,40)(30,33,37,34);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,30)(26,35)(27,28)(29,34)(31,37)(32,33)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=gh,e^2=g,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, 7772300);