#7865068

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 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)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,39,37);(1,3,5,8)(2,6,4,7)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(10,15)(11,16)(17,22)(18,23)(25,27,30,35)(26,31,33,28)(29,34,38,39)(32,37,40,36);(1,4,5,2)(3,7,8,6)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,27)(29,37)(30,31)(32,39)(33,35)(34,40)(36,38);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=gh,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, 7772264);