#118730

order	512
nilpotency class	3
isoclinism	isoclinic class of #118730
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,g^b=ghi,e^c=eh,f^c=fi

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, 25926);