#249900

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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1969 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)(11,20)(12,22)(13,16)(14,19)(15,18)(17,24)(21,23);(1,2)(3,5)(4,6)(7,8)(9,11,15,21)(10,13,18,12)(14,22,24,16)(17,23,19,20);(1,3)(2,5)(4,7)(6,8)(9,12,15,13)(10,16,18,22)(11,17,21,19)(14,23,24,20);(1,2)(3,5)(4,6)(7,8)(9,13,15,12)(10,11,18,21)(14,23,24,20)(16,17,22,19)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^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, 157096);