#502884

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

833 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)(4,5)(6,8)(9,10,13,19)(11,23,25,17)(12,28,31,59)(14,36,35,50)(15,22,16,21)(18,43,45,32)(20,38,48,30)(24,53,55,71)(26,57,37,69)(27,42,39,51)(29,62,63,58)(33,61,34,60)(40,56,52,70)(41,54,49,65)(44,67,72,64)(46,68,47,66);(2,5)(6,7)(9,11)(10,17)(12,29)(13,25)(14,37)(15,39)(16,27)(18,44)(19,23)(20,49)(21,51)(22,42)(24,33)(26,35)(28,58)(30,65)(31,63)(32,67)(34,55)(36,57)(38,54)(40,46)(41,48)(43,64)(45,72)(47,52)(50,69)(53,60)(56,66)(59,62)(61,71)(68,70);(9,12,30,66,16,34,35,43,13,31,38,68,15,33,14,32)(10,18,36,61,22,47,48,59,19,45,50,60,21,46,20,28)(11,24,54,64,27,29,37,70,25,55,65,67,39,63,26,56)(17,40,69,62,42,44,49,53,23,52,57,58,51,72,41,71);(1,3,4,8)(2,6,5,7)(10,19)(12,31)(17,23)(20,48)(21,22)(24,55)(29,63)(32,43)(33,34)(36,50)(41,49)(42,51)(56,70)(57,69)(64,67)(66,68)

Polycyclic presentation:

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

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, 410080);