#120881

order	512
nilpotency class	3
isoclinism	isoclinic class of #120880
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

453 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:

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

Polycyclic presentation:

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

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, 28077);