#254919

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59126	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1706 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16);(2,5)(6,8)(10,13)(12,16)(17,18)(19,21)(20,23)(22,24);(1,3,4,7)(2,6,5,8)(10,14)(12,16)(13,15)(18,21)(19,24)(22,23);(2,5)(6,8)(10,13)(14,15)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264256, #72264308, #72264312
maximal normal subgroup of	#72264256, #72264308, #72264312

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162115);