#249883

order	512
nilpotency class	3
isoclinism	isoclinic class of #249879
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60372	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1949 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157079);