#481846

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#36020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1034 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)(3,5)(4,6)(7,8)(9,10,13,18,14,17,25,31)(11,19,21,32,22,15,26,28)(12,16,24,30,23,29,36,39)(20,33,35,40,34,27,37,38);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,17,19)(12,20,23,34)(13,21,25,26)(16,27,29,33)(18,32,31,28)(24,35,36,37)(30,40,39,38);(2,6)(5,7)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,29)(18,30)(19,33)(21,35)(22,34)(25,36)(26,37)(28,38)(31,39)(32,40);(12,23)(16,29)(20,34)(24,36)(27,33)(30,39)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^e=fi

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, 389042);