#208072

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2428	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115268);