#6605592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2197 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,19,23,20,22,24,21);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(18,21)(19,24)(22,23);(2,6)(5,8)(10,14)(11,15)(18,22)(21,23);(9,11)(10,13)(12,15)(14,16)(18,22)(21,23);(11,15)(13,16)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305441, #72305454, #72305478
maximal normal subgroup of	#72305441, #72305454, #72305478
maximal quotient of	#72305925, #72306113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512788);