#149255

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1942
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

506 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)(5,6,10,19,13,20,12,17)(7,21,24,40,27,51,26,50)(8,28,31,15,33,46,36,47)(9,18,34,49,35,48,11,16)(14,37,41,56,42,25,39,23)(22,52,54,43,55,64,57,67)(29,59,60,38,61,65,53,66)(30,58,32,45,62,44,63,68);(1,2,3,4)(5,7,11,25,12,26,35,56,13,27,34,37,10,24,9,23)(6,14,18,40,19,41,49,51,20,42,48,50,17,39,16,21)(8,29,32,54,31,60,62,55,33,61,63,57,36,53,30,22)(15,43,45,59,28,52,58,66,47,67,68,65,46,64,44,38);(5,8,10,31,13,33,12,36)(6,15,17,28,20,47,19,46)(7,22,24,54,27,55,26,57)(9,32,34,62,35,63,11,30)(14,38,39,59,42,66,41,65)(16,45,48,58,49,68,18,44)(21,43,50,52,51,67,40,64)(23,29,37,60,56,61,25,53)

Polycyclic presentation:

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

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, 56451);