#484094

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

451 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234320, #72234327, #72234332, …
maximal normal subgroup of	#72234320, #72234327, #72234332, …
maximal quotient of	#72320193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391290);