#149128

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	#1934
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

420 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:

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

Polycyclic presentation:

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

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, 56324);