#120046

order	512
nilpotency class	3
isoclinism	isoclinic class of #99835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

610 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,e^2=hi,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=ehi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27242);