#498510

order	512
nilpotency class	4
isoclinism	isoclinic class of #484139
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33605	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

608 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ef,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264744, #72264749, #72264774, …
maximal normal subgroup of	#72264744, #72264749, #72264774, …
maximal quotient of	#72343504, #72343515, #72343569

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405706);