#149189

order	512
nilpotency class	4
isoclinism	isoclinic class of #134690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#106
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=ei,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,g^b=gi

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, 56385);