#189234

order	512
nilpotency class	3
isoclinism	isoclinic class of #19254
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96430);