#8188138

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	#37703	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4531 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,20)(13,15)(14,16)(17,22)(19,23)(21,24)(25,26)(27,28)(29,35)(30,32)(31,33)(34,36)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27)(26,29)(28,33)(30,34)(31,35)(32,37)(36,39)(38,40);(1,2,4,6)(3,5,8,7)(10,16)(15,18)(17,20)(23,24)(25,28,31,29)(26,27,33,35)(30,36,38,37)(32,34,39,40);(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,29,31,28)(26,27,33,35)(30,37,38,36)(32,34,39,40);(1,3)(2,5)(4,8)(6,7)(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,30,31,38)(26,32,33,39)(27,34,35,40)(28,36,29,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36)(37,39)(38,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095334);