#181116

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1962 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)(11,19,22,15)(12,16)(13,17)(14,18)(20,24,21,23);(2,6,8,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24);(1,3,5,4)(2,7,8,6)(9,13,14,12)(10,17,18,16)(11,21,22,20)(15,24,19,23)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211165, #72211189, #72211196, …
maximal normal subgroup of	#72211165, #72211189, #72211196, …
maximal quotient of	#72239228, #72239590

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88312);