#118731

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	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

766 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 25927);