#110181

order	512
nilpotency class	3
isoclinism	isoclinic class of #110180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

831 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:

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

Polycyclic presentation:

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

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, 17377);