#146930

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	#1934
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 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)(7,18)(8,24)(9,29)(10,32)(11,34)(12,17)(13,20)(14,26)(15,31)(16,30)(19,46)(21,37)(22,54)(23,45)(25,40)(27,61)(28,38)(33,41)(35,49)(36,53)(39,59)(42,48)(43,52)(44,51)(47,55)(50,63)(56,60)(57,58)(62,66)(64,67)(65,68);(5,7,10,21,30,51,31,52,12,23,9,20,11,22,33,53)(6,13,15,18,34,43,32,54,17,37,41,45,16,36,29,44)(8,19,26,48,58,66,59,67,28,50,25,47,27,49,60,68)(14,35,38,42,56,63,57,65,40,62,24,55,39,46,61,64);(5,8,9,25,30,58,33,60,12,28,10,26,11,27,31,59)(6,14,15,38,34,56,32,57,17,40,41,24,16,39,29,61)(7,19,20,47,51,66,53,68,23,50,21,48,22,49,52,67)(13,35,18,42,43,63,54,65,37,62,45,55,36,46,44,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,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, 54126);