#502874

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
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	#60682	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

842 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)(9,10,15,19)(11,16,22,37)(12,25,29,56)(13,31,34,64)(14,35,33,20)(17,40,42,28)(18,43,45,27)(21,46,26,53)(23,51,32,63)(24,52,50,39)(30,62,61,55)(36,58,65,48)(38,57,44,47)(41,60,54,67)(49,71,59,69)(66,70,68,72);(2,5)(6,7)(9,11)(10,16)(12,26)(13,32)(14,24)(15,22)(17,36)(18,44)(19,37)(20,39)(21,29)(23,34)(25,53)(27,57)(28,48)(30,59)(31,63)(33,50)(35,52)(38,45)(40,58)(41,66)(42,65)(43,47)(46,56)(49,61)(51,64)(54,68)(55,71)(60,70)(62,69)(67,72);(9,12,27,60,14,30,34,40,15,29,43,67,33,61,13,28)(10,17,31,55,20,41,45,56,19,42,64,62,35,54,18,25)(11,21,47,70,24,49,32,58,22,26,57,72,50,59,23,48)(16,36,51,69,39,66,44,53,37,65,63,71,52,68,38,46);(1,3,4,8)(2,6,5,7)(11,22)(16,37)(21,26)(23,32)(24,50)(36,65)(38,44)(39,52)(46,53)(47,57)(48,58)(49,59)(51,63)(66,68)(69,71)(70,72)

Polycyclic presentation:

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

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, 410070);