#489018

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61290	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 396214);