#147218

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1874
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54414);