#8095922

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	#45530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4459 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003118);