#149160

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2073
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

422 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)(7,20)(8,27)(9,16)(10,15)(11,37)(12,19)(13,22)(14,30)(17,36)(18,33)(21,53)(23,51)(24,42)(25,50)(26,52)(28,61)(29,63)(31,45)(32,48)(34,49)(35,47)(38,40)(39,57)(41,59)(43,55)(44,64)(46,65)(54,66)(56,60)(58,68)(62,67);(5,7)(6,13)(8,28,32,54)(9,23)(10,22)(11,38)(12,26)(14,43,47,68)(15,40)(16,20)(17,50)(18,51)(19,42)(21,29,57,65)(24,36)(25,33)(27,60,48,67)(30,62,35,56)(31,58,64,55)(34,59)(37,41)(39,45,53,44)(46,66,63,61)(49,52);(5,8,10,35,11,31,34,65,12,32,36,30,33,64,9,29)(6,14,16,48,18,46,17,45,19,47,49,27,37,63,15,44)(7,21,23,58,25,56,24,28,26,57,59,55,38,62,22,54)(13,39,40,61,41,60,52,68,42,53,50,66,51,67,20,43)

Polycyclic presentation:

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

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, 56356);