#114415

order	512
nilpotency class	3
isoclinism	isoclinic class of #114414
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

1004 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)(4,12,13,7)(5,8)(10,14)(15,16)(18,23,26,25)(24,27,31,32);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,15,12,14)(17,18)(19,27)(20,25)(21,23)(22,26)(24,28)(29,32)(30,31);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,27)(25,32)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21611);