#497029

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33604	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 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)(3,7)(4,6)(5,8)(9,10,14,20,16,21,15,36)(11,17,25,40,23,38,26,54)(12,27,31,57,29,55,32,60)(13,33,34,44,35,61,28,19)(18,41,43,59,42,58,56,30)(22,45,48,67,50,68,49,71)(24,52,53,65,51,72,46,39)(37,62,63,70,64,69,66,47);(1,3)(2,5)(4,8)(6,7)(9,11,14,25,16,23,15,26)(10,17,20,40,21,38,36,54)(12,22,31,48,29,50,32,49)(13,24,34,53,35,51,28,46)(18,37,43,63,42,64,56,66)(19,39,33,52,44,65,61,72)(27,45,57,67,55,68,60,71)(30,47,41,62,59,70,58,69);(3,8)(5,7)(9,12,28,58,15,32,35,59,16,29,34,41,14,31,13,30)(10,18,33,57,20,43,44,55,21,42,61,60,36,56,19,27)(11,22,46,69,26,49,51,70,23,50,53,62,25,48,24,47)(17,37,52,67,40,63,65,68,38,64,72,71,54,66,39,45);(2,6)(5,7)(11,23)(12,29)(17,38)(18,42)(24,51)(25,26)(27,55)(30,59)(31,32)(39,65)(40,54)(41,58)(43,56)(46,53)(52,72)(57,60)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404225);