#147060

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1804
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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)(7,17)(8,22,24,13)(9,26)(10,32)(11,15)(12,19)(14,30)(16,29)(18,43,46,64)(20,44)(21,50)(23,53,28,62)(25,38,56,52)(27,42)(31,59)(33,39)(34,45,51,48)(35,61)(36,49)(37,57,40,63)(41,55)(47,66,58,68)(54,65,60,67);(1,2,3,4)(5,7)(6,12)(8,18)(9,27)(10,33)(11,20)(13,34)(14,39)(15,36)(16,42)(17,26)(19,30)(21,29)(22,51)(23,54)(24,46)(25,58)(28,60)(31,61)(32,35)(37,66)(38,67)(40,68)(41,50)(43,53)(44,59)(45,57)(47,56)(48,63)(49,55)(52,65)(62,64);(5,8,10,25,11,24,29,56)(6,13,32,52,15,22,16,38)(7,18,33,58,20,46,21,47)(9,28,30,63,31,23,55,57)(12,34,35,65,36,51,42,67)(14,40,59,62,41,37,26,53)(17,43,39,68,44,64,50,66)(19,48,61,54,49,45,27,60)

Polycyclic presentation:

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

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, 54256);