#251726

order	512
nilpotency class	3
isoclinism	isoclinic class of #251701
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

Polycyclic presentation:

2^9|b^2=ehi,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^c=di,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, 158922);