#497933

order	512
nilpotency class	4
isoclinism	isoclinic class of #483463
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂²×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

474 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309514, #72309518, #72309521, …
maximal normal subgroup of	#72309514, #72309518, #72309521, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405129);