#248651

order	512
nilpotency class	3
isoclinism	isoclinic class of #248647
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	#36078	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1958 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)(9,10,14,12)(11,17,20,24)(13,21,23,16)(15,22,18,19);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,22)(14,20)(16,24)(17,21)(19,23);(5,8)(9,12,14,10)(11,18,20,15)(13,16,23,21)(17,22,24,19);(9,11)(10,15)(12,18)(13,22)(14,20)(16,24)(17,21)(19,23)

Polycyclic presentation:

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

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, 155847);