#147487

order	512
nilpotency class	4
isoclinism	isoclinic class of #13030
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

708 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)(5,6,10,16,11,15,24,17)(7,18,21,46,23,45,22,40)(8,13,27,38,26,39,28,43)(9,29,31,14,32,42,53,41)(12,30,35,52,34,51,36,20)(19,44,49,63,48,67,50,66)(25,54,57,64,56,65,58,37)(33,59,60,68,62,55,61,47);(5,7,10,21,11,23,24,22)(6,12,17,36,15,34,16,35)(8,19,28,50,26,48,27,49)(9,30,31,52,32,51,53,20)(13,33,38,60,39,62,43,61)(14,40,29,45,41,46,42,18)(25,55,58,68,56,59,57,47)(37,63,54,67,64,66,65,44);(1,2,3,4)(5,8,24,27,11,26,10,28)(6,13,16,43,15,39,17,38)(7,19,22,49,23,48,21,50)(9,25,53,57,32,56,31,58)(12,33,35,61,34,62,36,60)(14,37,42,65,41,64,29,54)(18,44,46,66,45,67,40,63)(20,47,51,59,52,68,30,55)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54683);