#319869

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 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)(4,6)(5,8)(10,13)(15,16)(17,18,20,24,22,23,21,19);(1,3,4,8)(2,5,6,7)(9,10,14,15,11,13,12,16)(17,19,21,23,22,24,20,18);(10,15)(12,14)(13,16)(18,19)(20,21)(23,24);(2,6)(5,7)(18,23)(19,24)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226827
maximal normal subgroup of	#72226827
maximal quotient of	#72224935

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227065);