#118749

order	512
nilpotency class	3
isoclinism	isoclinic class of #118747
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

791 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,15,16,14)(17,18,21,23,22,26,20,25)(19,24,29,30,27,32,28,31);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,14)(13,16)(18,23)(19,27)(20,21)(24,31)(25,26)(30,32);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(17,19,20,28,22,27,21,29)(18,24,25,31,26,32,23,30)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,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, 25945);