#120369

order	512
nilpotency class	3
isoclinism	isoclinic class of #120369
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
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

454 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fi,f^c=fi,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, 27565);