#8131213

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4256 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8038409);