#508178

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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	#13307	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

819 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,27)(13,19)(14,34)(15,21)(16,39)(17,24)(18,30)(20,37)(23,52)(25,56)(26,43)(28,62)(29,61)(31,48)(32,65)(33,66)(35,49)(36,47)(38,42)(40,68)(41,69)(44,72)(45,63)(46,64)(50,54)(51,58)(53,70)(55,67)(57,71)(59,60);(2,5)(6,8)(9,11)(10,17)(12,28)(13,24)(14,25)(15,38)(16,26)(18,44)(19,22)(20,41)(21,43)(23,31)(27,60)(29,59)(30,57)(32,53)(33,55)(34,67)(35,68)(36,69)(37,58)(39,42)(40,46)(45,71)(47,51)(48,70)(49,50)(52,65)(54,64)(56,66)(61,62)(63,72);(3,7)(6,8)(9,12)(10,18)(11,23)(13,29)(14,35)(15,32)(16,31)(17,40)(19,45)(20,48)(21,49)(22,50)(24,53)(25,57)(26,59)(27,34)(28,38)(30,37)(33,64)(36,63)(39,46)(41,62)(42,71)(43,44)(47,65)(51,60)(52,56)(54,58)(55,72)(61,66)(67,70)(68,69);(1,3,4,7)(2,6,5,8)(12,29)(18,45)(23,53)(27,61)(28,59)(30,63)(31,32)(35,64)(40,50)(44,71)(46,49)(48,65)(52,70)(54,68)(57,72)(60,62)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415374);