#95584

order	512
nilpotency class	3
isoclinism	isoclinic class of #94966
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gh,e^2=i,f^2=hi,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,e^b=eh

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, 2780);