#8195373

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	#36518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,e^2=h,f^2=ghi,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, 8102569);