#8702703

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	#47676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8609899);