#497642

order	512
nilpotency class	4
isoclinism	isoclinic class of #480866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#554 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,d^2=ei,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,g^c=gh,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404838);