#147840

order	512
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

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, 55036);