#182097

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,f^2=i,g^2=h,b^a=be,g^a=gh,e^b=eh,d^c=dh

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89293);