#449777

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356973);