#183563

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

812 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=i,b^a=be,d^a=dh,c^b=ci,e^b=eh,f^b=fh,d^c=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, 90759);