#117830

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

748 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,d^c=di,f^c=fhi

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, 25026);