#248477

order	512
nilpotency class	3
isoclinism	isoclinic class of #248455
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	C₂³×C₄	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155673);