#129543

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂×#121	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dgi,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36739);