#483802

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×#508	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

469 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)(9,10,14,17,15,20,12,18)(11,21,22,38,25,39,24,36)(13,27,30,37,29,35,26,19)(16,31,33,23,34,28,32,40);(9,11)(10,16)(12,22)(13,28)(14,24)(15,25)(17,32)(18,33)(19,36)(20,34)(21,35)(23,30)(26,40)(27,39)(29,31)(37,38);(1,3,4,6,5,7,8,2)(10,17,20,18)(13,29)(16,32,34,33)(19,35,37,27)(21,38,39,36)(23,40)(26,30)(28,31);(9,12,15,14)(10,18,20,17)(11,22,25,24)(13,26,29,30)(16,33,34,32)(19,35,37,27)(21,38,39,36)(23,28,40,31)

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,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	#72285482, #72285766, #72286241, …
maximal normal subgroup of	#72285482, #72285766, #72286241, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390998);