#208576

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

916 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=f,e^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115772);