#6453848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437445
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3285 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)(9,10)(11,21)(12,18)(13,19)(14,16)(15,20)(17,23)(22,24);(10,16)(12,22)(13,14)(17,18)(19,20)(21,24);(2,5)(6,8)(9,11,15,23)(10,17,20,21)(12,13,22,14)(16,24,19,18);(9,12)(10,18)(11,13)(14,23)(15,22)(16,17)(19,21)(20,24);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,19,20,16)(11,22,23,12)(17,18,21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257430, #72280219, #72319159, …
maximal normal subgroup of	#72257430, #72280219, #72319159, …
maximal quotient of	#72264510

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361044);