#8746033

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	#40525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653229);