#208320

order	512
nilpotency class	3
isoclinism	isoclinic class of #162342
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

753 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)(11,18)(12,16)(13,22)(14,17)(15,19)(20,23)(21,24)(25,26,29,31,30,34,28,33)(27,32,37,38,35,39,36,40);(9,11,13,19,14,20,12,21)(10,15,16,18,17,24,22,23)(26,31)(28,29)(32,38)(33,34)(36,37)(39,40);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18)(25,27,30,35)(26,32,34,39)(28,36,29,37)(31,38,33,40);(1,3,4,7)(2,6,5,8)(11,20)(15,24)(18,23)(19,21)(27,35)(32,39)(36,37)(38,40)

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,c^b=ch,d^b=dh,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, 115516);