#147124

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×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	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

540 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)(7,18)(8,24)(9,29)(10,16)(11,15)(12,17)(13,20)(14,26)(19,45)(21,36)(22,44)(23,34)(25,39)(27,53)(28,37)(30,42)(31,40)(32,41)(33,48)(35,49)(38,55)(43,46)(47,64)(50,58)(51,60)(52,59)(54,62)(56,61)(57,66)(63,65)(67,68);(5,7,10,22,11,20,31,49,12,23,32,52,9,21,30,51)(6,13,16,35,15,34,40,59,17,36,41,60,29,18,42,44)(8,19,27,47,26,46,55,65,28,48,56,66,25,50,54,67)(14,33,38,57,37,58,61,68,39,45,62,64,24,43,53,63);(1,3,4,2)(5,8)(6,14)(7,19)(9,25)(10,27)(11,26)(12,28)(13,33)(15,37)(16,38)(17,39)(18,43)(20,46)(21,50)(22,47)(23,48)(24,29)(30,54)(31,55)(32,56)(34,58)(35,57)(36,45)(40,61)(41,62)(42,53)(44,63)(49,65)(51,67)(52,66)(59,68)(60,64)

Polycyclic presentation:

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

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, 54320);