#8108019

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 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:

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

Polycyclic presentation:

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

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, 8015215);