#253429

order	512
nilpotency class	3
isoclinism	isoclinic class of #253411
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	C₂³×C₄	C₂⁴
prime radical	#18316	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1006 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)(11,17)(12,20)(14,23)(15,21)(18,24)(19,22);(2,5)(4,8)(9,10)(11,18,19,14)(12,21)(13,16)(15,20)(17,23,22,24);(2,6)(4,8)(5,7)(9,11)(10,14)(12,22)(13,19)(15,24)(16,18)(17,20)(21,23);(2,5)(6,7)(12,20)(15,21)(17,22)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160625);