#120948

order	512
nilpotency class	3
isoclinism	isoclinic class of #120908
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

432 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:

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

Polycyclic presentation:

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

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, 28144);