#8745953

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	#41944	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=hi,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, 8653149);