#488917

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61088	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

973 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)(4,7)(5,8)(9,10,15,20)(11,21,24,50)(12,17,26,41)(13,29,34,64)(14,35,32,19)(16,37,39,23)(18,44,47,33)(22,51,52,49)(25,56,36,45)(27,58,31,63)(28,61,57,43)(30,48,66,40)(38,55,68,53)(42,67,46,60)(54,72,62,70)(59,71,65,69);(2,6)(3,4)(7,8)(9,11)(10,16)(12,22)(13,30)(14,36)(15,24)(17,38)(18,45)(19,48)(20,39)(21,29)(23,33)(25,32)(26,52)(27,59)(28,62)(31,65)(34,66)(35,40)(37,44)(41,68)(42,70)(43,71)(46,72)(47,56)(49,63)(50,64)(51,58)(53,60)(54,57)(55,67)(61,69);(2,7)(3,4)(6,8)(9,12)(10,17)(11,22)(13,31)(14,28)(15,26)(16,38)(18,46)(19,43)(20,41)(21,49)(23,55)(24,52)(25,54)(27,34)(29,63)(30,65)(32,57)(33,67)(35,61)(36,62)(37,53)(39,68)(40,69)(42,47)(44,60)(45,72)(48,71)(50,51)(56,70)(58,64)(59,66);(12,26)(17,41)(22,52)(27,31)(28,57)(38,68)(42,46)(43,61)(49,51)(53,55)(54,62)(58,63)(59,65)(60,67)(69,71)(70,72)

Polycyclic presentation:

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

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, 396113);