#6470566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35077	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2219 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)(17,18)(19,22)(20,23)(21,24);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,20)(18,22,24,23);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,23)(19,20)(22,24);(11,15)(13,16)(17,19,21,20)(18,22,24,23);(10,14)(13,16)(18,24)(22,23)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305463, #72305470, #72305478
maximal normal subgroup of	#72305463, #72305470, #72305478
maximal quotient of	#72305877, #72306113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377762);