#484211

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1083 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

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, 391407);