#111648

order	512
nilpotency class	3
isoclinism	isoclinic class of #96944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18844);