#8135392

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	#42997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4688 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8042588);