#7947058

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	#44651	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7854254);