#8752440

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	#48057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4397 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8659636);