#252235

order	512
nilpotency class	3
isoclinism	isoclinic class of #252235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2404
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1107 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)(4,7)(6,8)(10,14)(16,23);(9,10,12,14)(11,17)(13,16,20,23)(15,21)(18,22)(19,24);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,22)(16,24)(17,23)(18,20);(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279995, #72280510, #72280571
maximal normal subgroup of	#72279995, #72280510, #72280571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159431);