#96477

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

590 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)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,11,13,21,14,22,12,20)(10,15,16,23,18,24,17,19)(25,27)(26,30,32,33)(28,36)(29,35)(31,39,40,38)(34,37);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=ch,d^b=dh

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, 3673);