#114328

order	512
nilpotency class	3
isoclinism	isoclinic class of #114327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=dhi,e^c=ei,f^c=fh,g^c=gh

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, 21524);