#250227

order	512
nilpotency class	3
isoclinism	isoclinic class of #250199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,e^c=ei,g^c=gi,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, 157423);