#120977

order	512
nilpotency class	3
isoclinism	isoclinic class of #120974
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

423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28173);