#267407

order	512
nilpotency class	3
isoclinism	isoclinic class of #267371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61564	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1147 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)(3,7,8,5)(4,6)(11,18)(15,23)(19,24)(20,22);(1,3)(2,5)(4,8)(6,7)(9,10)(11,19)(12,16)(13,17)(14,21)(15,20)(18,24)(22,23);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,18)(14,22)(16,24)(17,23)(19,21);(1,3)(2,5)(4,8)(6,7)(9,12,13,14)(10,16,17,21)(11,20,18,22)(15,24,23,19)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174603);