#147416

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^2=i,g^2=hi,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, 54612);