#111138

order	512
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(2,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,19,24,31)(16,20)(17,28)(18,21)(22,29)(23,35,36,30)(25,39,38,32)(26,34)(27,41)(33,40)(37,42,44,43);(6,9)(7,12)(10,11)(13,15,17,25,18,24,29,38)(14,19,22,32,21,31,28,39)(16,23,27,37,26,36,40,44)(20,30,33,42,34,35,41,43);(1,2,3,4)(13,16,18,26)(14,20,21,34)(15,23,24,36)(17,27,29,40)(19,30,31,35)(22,33,28,41)(25,37,38,44)(32,42,39,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18334);