#147916

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂²×#508	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

962 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)(5,6,10,17,12,18,11,34)(7,19,22,50,23,38,35,48)(8,14,26,45,27,47,36,46)(9,28,31,16,33,52,32,15)(13,29,39,58,41,63,40,21)(20,53,54,42,56,67,55,65)(24,59,60,66,61,68,57,37)(25,49,30,43,62,51,64,44);(5,7)(6,13)(8,24)(9,29)(10,22)(11,35)(12,23)(14,42)(15,48)(16,50)(17,39)(18,41)(19,28)(20,25)(21,32)(26,60)(27,61)(30,54)(31,58)(33,63)(34,40)(36,57)(37,44)(38,52)(43,66)(45,67)(46,53)(47,65)(49,59)(51,68)(55,64)(56,62);(2,4)(5,8)(6,14)(7,20)(9,30)(10,26)(11,36)(12,27)(13,37)(15,49)(16,51)(17,45)(18,47)(19,53)(21,57)(22,54)(23,56)(24,29)(25,32)(28,43)(31,62)(33,64)(34,46)(35,55)(38,67)(39,59)(40,68)(41,66)(42,50)(44,52)(48,65)(58,60)(61,63)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,f^b=fh,h^b=hi,d^c=dh,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55112);