#146435

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

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	#1808
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

872 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)(8,26,29,45)(9,17)(10,16)(11,15)(12,19)(13,22)(14,28,41,31)(18,32)(21,49,52,67)(23,47)(24,36)(25,48)(27,59,61,40)(30,42,60,65)(33,43)(34,44)(35,62,63,53)(37,55)(38,57)(39,56)(46,51,66,54)(50,58,68,64);(2,4)(5,7,11,24,10,23,33,56,12,25,34,57,32,55,9,22)(6,13,17,37,18,38,44,48,19,39,43,47,16,36,15,20)(8,21,30,53,31,54,61,68,29,52,60,62,28,51,27,50)(14,35,42,49,26,58,59,66,41,63,65,67,45,64,40,46);(5,8)(6,14)(7,21)(9,27)(10,31)(11,30)(12,29)(13,35)(15,40)(16,45)(17,42)(18,26)(19,41)(20,46)(22,50)(23,54)(24,53)(25,52)(28,32)(33,61)(34,60)(36,64)(37,49)(38,58)(39,63)(43,65)(44,59)(47,67)(48,66)(51,55)(56,68)(57,62)

Polycyclic presentation:

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

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, 53631);