#7919668

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	#35539	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4416 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7826864);