#265339

order	512
nilpotency class	3
isoclinism	isoclinic class of #265339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18228	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1073 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)(11,17)(12,23)(13,22)(14,20)(15,19)(16,21)(18,24);(1,3)(2,5)(4,7)(6,8)(9,11,14,21,15,12,24,22)(10,16,18,17,19,13,20,23);(5,8)(9,12)(10,17)(11,15)(13,18)(14,22)(16,20)(19,23)(21,24);(9,13,14,23,15,16,24,17)(10,12,18,22,19,11,20,21)

Polycyclic presentation:

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

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, 172535);