#119579

order	512
nilpotency class	3
isoclinism	isoclinic class of #119578
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

715 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)(3,11)(4,8)(5,15)(6,10)(7,12)(9,14)(13,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,32,31)(26,30);(1,3,6,13)(2,7,10,8)(4,14,12,5)(9,16,15,11)(17,19)(18,22)(20,25)(21,26)(23,31)(24,30)(27,29)(28,32);(1,4,5,3,6,12,14,13)(2,8,9,16,10,7,15,11)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,e^c=eh,g^c=gi

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, 26775);