#181240

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#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 ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1810 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214439, #72214831, #72214857, …
maximal normal subgroup of	#72214439, #72214831, #72214857, …
maximal quotient of	#72240809

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88436);