#146786

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,4)(10,15,16,24)(11,18)(12,22)(13,21)(17,19,23,20);(2,4)(5,6,7,8)(9,10)(11,19)(12,20)(13,23)(14,16)(15,22)(17,21)(18,24);(9,10)(11,20)(12,19)(13,23)(14,16)(15,18)(17,21)(22,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242475, C₄×#13025, #72242479
maximal normal subgroup of	#72242475, C₄×#13025, #72242479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53982);