#8125458

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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)(3,7)(9,10,15,12)(11,16,18,13)(14,20,22,17)(19,21,24,23)(26,29)(27,33)(32,37)(34,39);(1,2,4,5)(3,6,7,8)(9,11,15,18)(10,16,12,13)(14,19,22,24)(17,23,20,21)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,2)(3,6)(4,5)(7,8)(11,18)(13,16)(19,24)(21,23)(28,36)(32,37)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,28,30,36)(26,32,29,37)(27,34,33,39)(31,38,35,40);(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)(25,29,30,26)(27,35,33,31)(28,37,36,32)(34,40,39,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8032654);