#147187

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1893
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

520 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)(9,30)(10,33)(11,35)(12,17)(13,20)(14,26)(15,32)(16,31)(19,47)(21,55)(22,38)(23,46)(25,39)(27,61)(28,41)(29,40)(34,42)(36,63)(37,56)(43,49)(44,54)(45,53)(48,57)(50,64)(51,67)(52,66)(58,60)(59,62)(65,68);(5,7,11,22,31,53,32,54,12,23,9,20,10,21,34,56)(6,13,15,18,33,44,35,55,17,38,42,46,16,37,30,45)(8,19,27,50,26,49,60,68,29,52,25,48,28,51,59,63)(14,36,39,43,24,57,58,47,41,65,61,67,40,64,62,66);(1,3,4,2)(5,8)(6,14)(7,19)(9,25)(10,28)(11,27)(12,29)(13,36)(15,39)(16,40)(17,41)(18,43)(20,48)(21,51)(22,50)(23,52)(24,33)(26,31)(30,62)(32,60)(34,59)(35,58)(37,64)(38,65)(42,61)(44,57)(45,66)(46,67)(47,55)(49,53)(54,68)(56,63)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54383);