#495632

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1033 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)(3,6)(4,5)(7,8)(9,10,14,18,13,17,26,28)(11,19,22,34,21,15,27,30)(12,23,25,37,24,16,31,32)(20,29,36,39,35,33,38,40);(9,11,14,27)(10,15,28,34)(12,20,25,38)(13,21,26,22)(16,29,37,40)(17,19,18,30)(23,33,32,39)(24,35,31,36);(2,5)(6,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,29)(17,23)(18,32)(19,33)(21,35)(22,36)(26,31)(27,38)(28,37)(30,39)(34,40);(1,3,4,7)(2,6,5,8)(12,24)(16,23)(20,35)(25,31)(29,33)(32,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fi,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402828);