#8115757

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	#34988	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022953);