#496236

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1764 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403432);