#120379

order	512
nilpotency class	3
isoclinism	isoclinic class of #120378
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

465 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:

(2,6)(3,10)(4,7)(8,15)(9,13)(12,16)(17,18)(19,25,28,22)(20,29)(21,23)(24,31)(26,30,32,27);(1,2,7,13,5,9,4,6)(3,8,14,15,10,16,11,12)(17,18)(19,26)(20,29)(21,23)(22,27)(24,31)(25,30)(28,32);(1,3,4,11,5,10,7,14)(2,8,6,12,9,16,13,15)(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=dh,c^2=d,d^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,e^c=eh,f^c=fi,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, 27575);