#110122

order	512
nilpotency class	3
isoclinism	isoclinic class of #110112
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

787 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)(4,7,12,14)(5,8)(10,13)(15,16)(17,18)(19,24,28,25)(20,30,29,32)(21,27)(22,31)(23,26);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,29,23,28,21,20)(18,24,27,30,26,25,31,32);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(17,20,21,28,23,29,22,19)(18,25,27,32,26,24,31,30)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,g^2=i,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=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17318);