#496006

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
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	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 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)(3,8)(4,6)(5,7)(11,19,21,37)(12,23)(13,26)(15,27,29,39)(17,32)(18,31)(20,36,22,24)(25,35)(28,33,34,40)(30,38);(1,3,5,4)(2,6,7,8)(9,10,14,16)(11,15,21,29)(12,17,25,30)(13,18,26,31)(19,27,37,39)(20,34,22,28)(23,32,35,38)(24,33,36,40);(9,11)(10,15)(12,24)(13,20)(14,21)(16,29)(17,33)(18,34)(19,35)(22,26)(23,37)(25,36)(27,38)(28,31)(30,40)(32,39);(1,4,5,3)(2,6,7,8)(10,16)(11,20,21,22)(12,25)(15,28,29,34)(18,31)(19,36,37,24)(23,35)(27,33,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=eg,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343219, #72343238, #72343260
maximal normal subgroup of	#72343219, #72343238, #72343260
maximal quotient of	#72343321, #72343390

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403202);