#502944

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

731 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410140);