#121056

order	512
nilpotency class	3
isoclinism	isoclinic class of #121054
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

431 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:

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

Polycyclic presentation:

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

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, 28252);