#107534

order	512
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

776 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)(37,38,42,47,43,48,41,46)(39,49,53,44,52,59,51,58)(40,45,54,60,55,61,56,62)(50,64,65,68,66,63,67,57);(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)(38,44)(40,50)(41,53)(42,51)(43,52)(45,57)(46,59)(47,49)(48,58)(54,67)(55,66)(56,65)(60,63)(61,68)(62,64);(5,8)(6,12)(7,16)(9,24)(10,20)(11,26)(13,28)(14,30)(15,31)(17,35)(18,33)(19,22)(21,25)(23,34)(27,32)(29,36)(37,40)(38,45)(39,50)(41,56)(42,54)(43,55)(44,57)(46,62)(47,60)(48,61)(49,63)(51,67)(52,66)(53,65)(58,68)(59,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh

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, 14730);