#495495

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3455	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,21,35,48,16,22,36,49,13,19,24,38)(11,23,27,51,26,41,52,47,28,46,34,61,33,17,37,40)(12,18,31,44,32,45,60,69,29,42,57,66,30,43,53,63)(25,50,56,70,55,64,71,68,54,67,59,72,58,39,62,65);(9,11,24,52,13,33,36,27,16,28,35,37,15,26,14,34)(10,17,38,51,19,46,49,40,22,41,48,61,21,23,20,47)(12,25,53,71,30,58,57,56,29,54,60,62,32,55,31,59)(18,39,63,70,43,67,66,65,42,64,69,72,45,50,44,68);(2,5)(6,8)(9,12)(10,18)(11,25)(13,30)(14,31)(15,32)(16,29)(17,39)(19,43)(20,44)(21,45)(22,42)(23,50)(24,53)(26,55)(27,56)(28,54)(33,58)(34,59)(35,60)(36,57)(37,62)(38,63)(40,65)(41,64)(46,67)(47,68)(48,69)(49,66)(51,70)(52,71)(61,72);(1,3,4,7)(2,6,5,8)(12,29)(18,42)(25,54)(30,32)(31,57)(39,64)(43,45)(44,66)(50,67)(53,60)(55,58)(56,59)(62,71)(63,69)(65,68)(70,72)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402691);