#94941

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,24,21,23,29,31)(19,25,27,32,26,22,28,30)(33,34)(35,38)(36,40)(37,39)(41,45)(42,44)(43,46)(47,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,22)(20,28)(21,26)(23,25)(24,32)(27,29)(30,31)(33,35,36,41,37,42,43,47)(34,38,40,45,39,44,46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,23)(19,26)(24,31)(27,28)(34,39)(38,44)(40,46)(45,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,e^2=h,f^2=i,b^a=bd,c^a=chi,d^a=dg,c^b=ch,e^b=eg

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, 2137);