#147754

order	512
nilpotency class	4
isoclinism	isoclinic class of #147752
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10991
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,g^b=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267969, #72267996, #72267997, …
maximal normal subgroup of	#72267969, #72267996, #72267997, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54950);