#272414

order	512
nilpotency class	3
isoclinism	isoclinic class of #272411
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

930 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:

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

Polycyclic presentation:

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

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, 179610);