#484081

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×#509	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dh,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	#72260298, #72260367, #72260372
maximal normal subgroup of	#72260298, #72260367, #72260372

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391277);