#118536

order	512
nilpotency class	3
isoclinism	isoclinic class of #10464
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

650 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)(7,12)(8,11)(9,10)(13,14,17,22,19,16,18,21)(15,23,24,20,26,28,25,27);(5,6,9,12,8,11,7,10)(13,15,18,25,19,26,17,24)(14,20,22,28,16,27,21,23);(2,4)(5,6,8,11)(7,12,9,10)(13,16,19,14)(15,23,26,28)(17,22,18,21)(20,24,27,25)

Polycyclic presentation:

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

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, 25732);