#95037

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

548 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=i,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,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, 2233);