#114429

order	512
nilpotency class	3
isoclinism	isoclinic class of #114428
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1109 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,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, 21625);