#382944

order	512
nilpotency class	3
isoclinism	isoclinic class of #188766
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#383
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,c^b=cg,h^b=hi,d^c=de,f^c=fi

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, 290140);