#496343

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403539);