#496727

order	512
nilpotency class	4
isoclinism	isoclinic class of #483449
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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)(3,7)(9,10,12,16)(11,19)(13,26,24,39)(14,28,25,18)(15,29)(17,32,31,27)(20,22)(21,33)(23,38)(30,37)(34,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,16)(13,17)(14,18)(15,21)(19,29)(22,33)(23,35)(24,31)(25,28)(26,27)(30,36)(32,39)(34,40)(37,38);(2,5)(6,8)(9,11,13,22,14,23,27,37,12,21,24,29,25,36,32,40)(10,15,17,19,18,30,26,34,16,20,31,33,28,35,39,38);(1,3,4,7)(2,6,5,8)(10,16)(15,20)(17,31)(18,28)(19,33)(26,39)(30,35)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343250, #72343253, #72343266
maximal normal subgroup of	#72343250, #72343253, #72343266
maximal quotient of	#72343747

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403923);