#112372

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

778 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)(7,15,18,11)(8,19)(9,22)(10,13)(12,21)(14,25)(16,32,33,36)(17,27,23,29)(20,30)(24,28)(26,34,31,35)(37,38,42,48,43,47,41,46)(39,49,52,44,53,60,51,59)(40,54,55,62,56,45,57,61)(50,64,65,58,66,63,67,68);(1,2)(3,4)(5,7)(6,11)(8,16)(9,23)(10,18)(12,26)(13,15)(14,29)(17,25)(19,32)(20,33)(21,35)(22,27)(24,34)(28,31)(30,36)(37,39,43,53)(38,44,47,59)(40,50,56,66)(41,52,42,51)(45,58,54,68)(46,60,48,49)(55,67,57,65)(61,64,62,63);(2,4)(5,8,10,20)(6,12,13,28)(7,16,18,33)(9,24,25,21)(11,26,15,31)(14,30,22,19)(17,35,23,34)(27,32,29,36)(37,40)(38,45)(39,50)(41,55)(42,57)(43,56)(44,58)(46,61)(47,54)(48,62)(49,63)(51,65)(52,67)(53,66)(59,68)(60,64)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19568);