#251695

order	512
nilpotency class	3
isoclinism	isoclinic class of #251671
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18300	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=ehi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,e^c=eh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158891);