#6460668

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|c^2=ghi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343135, #72343152, #72343186
maximal normal subgroup of	#72343135, #72343152, #72343186
maximal quotient of	#72343635, #72343663

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367864);