#6446711

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4188 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,23)(22,24);(2,5)(6,7)(9,11,14,23,15,12,13,22)(10,16,18,21,20,24,19,17);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,15)(13,23)(14,22)(16,18)(19,24)(20,21);(9,12)(10,17)(11,15)(13,23)(14,22)(16,18)(19,24)(20,21);(9,13,15,14)(10,18,20,19)(11,22,12,23)(16,21,24,17)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267112
maximal normal subgroup of	#72267112
maximal quotient of	#72267110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353907);