#8172727

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^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,d^c=dgi,e^d=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, 8079923);