#6446732

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
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
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4154 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,21)(13,18)(15,19)(16,20)(17,24)(22,23);(2,5)(6,7)(9,11,14,10,16,19,15,20)(12,22,13,17,24,18,23,21);(1,3,4,8)(2,6,5,7)(9,12,16,24)(10,17,20,21)(11,18,19,22)(13,14,23,15);(9,13)(10,18)(11,21)(12,14)(15,24)(16,23)(17,19)(20,22);(9,14,16,15)(10,19,20,11)(12,23,24,13)(17,22,21,18)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257908, #72302234, #72312145, …
maximal normal subgroup of	#72257908, #72302234, #72312145, …
maximal quotient of	#72264524, #72302322

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353928);