#8108010

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

4318 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)(4,5)(6,8)(9,10,14,18)(11,15,16,12)(13,21,22,17)(19,20,24,23)(25,26,30,34)(27,35,38,31)(28,33,29,32)(36,39,37,40);(2,5)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,21)(22,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,40)(37,39);(1,3,4,8)(2,6,5,7)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,27,30,38)(26,31,34,35)(28,36,29,37)(32,39,33,40);(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(9,13,14,22)(10,17,18,21)(11,19,16,24)(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,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, 8015206);