#126946

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

698 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=dg,c^b=ci,d^b=di,e^b=egi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34142);