#483805

order	512
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#442
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287707, #72287713, #72287755, …
maximal normal subgroup of	#72287707, #72287713, #72287755, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391001);