#208172

order	512
nilpotency class	3
isoclinism	isoclinic class of #162406
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

761 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)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24)(25,26,29,33,30,34,28,32)(27,35,36,31,38,39,37,40);(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,35)(34,40);(2,5)(6,8)(9,11,13,17)(10,15,16,22)(12,18,19,23)(14,20,21,24)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,40,35);(1,3,4,7)(2,6,5,8)(11,17)(15,22)(18,23)(20,24)

Polycyclic presentation:

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

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, 115368);