#147368

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54564);