#502880

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
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	#13319	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

840 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)(11,23)(12,28)(13,19)(14,36)(15,22)(16,21)(17,25)(18,32)(20,30)(24,52)(26,56)(27,42)(29,61)(31,58)(33,60)(34,59)(35,49)(37,69)(38,47)(39,50)(40,70)(41,64)(43,63)(44,66)(45,65)(46,68)(48,53)(51,55)(54,71)(57,62)(67,72);(2,5)(6,7)(9,11)(10,17)(12,29)(13,25)(14,37)(15,39)(16,27)(18,43)(19,23)(20,48)(21,50)(22,42)(24,33)(26,35)(28,57)(30,64)(31,62)(32,67)(34,54)(36,56)(38,53)(40,45)(41,47)(44,72)(46,51)(49,69)(52,59)(55,65)(58,61)(60,71)(63,66)(68,70);(9,12,30,65,16,34,35,66,13,31,38,68,15,33,14,32)(10,18,36,60,22,46,47,58,19,44,49,59,21,45,20,28)(11,24,53,63,27,29,37,70,25,54,64,67,39,62,26,55)(17,40,69,61,42,43,48,52,23,51,56,57,50,72,41,71);(1,3,4,8)(2,6,5,7)(10,19)(17,23)(18,44)(20,47)(21,22)(28,58)(36,49)(40,51)(41,48)(42,50)(43,72)(45,46)(52,71)(56,69)(57,61)(59,60)

Polycyclic presentation:

2^9|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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410076);