#111669

order	512
nilpotency class	3
isoclinism	isoclinic class of #97642
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#35189	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

(6,9,10,11)(13,14)(15,22,24,18)(16,21)(17,20)(19,23,27,25)(26,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,23)(17,24)(19,21)(20,22)(25,26)(27,28);(2,4)(13,15)(14,19)(16,25)(17,24)(18,21)(20,27)(22,28)(23,26)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,g^b=ghi,e^c=eh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18865);