#120968

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(2,6)(3,9)(4,11)(7,14)(8,12)(13,15)(17,18,22,25,20,24,21,30)(19,23,28,31,29,32,27,26);(1,2)(3,7,9,15)(4,12)(5,8)(6,11)(10,14,16,13)(17,19,21,27,20,29,22,28)(18,23,25,31,24,32,30,26);(1,3,4,10,5,9,11,16)(2,7,6,13,8,15,12,14)(17,18,20,24)(19,26)(21,30,22,25)(23,28)(27,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28164);