#6449091

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263759, #72263763, #72263764
maximal normal subgroup of	#72263759, #72263763, #72263764
maximal quotient of	#72280101

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356287);