#181281

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88477);