#114221

order	512
nilpotency class	3
isoclinism	isoclinic class of #114220
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

758 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)(4,8)(5,18)(6,9)(10,21)(12,22,30,29)(14,23,19,27)(15,32,31,24)(16,28)(17,25)(20,26)(33,34)(35,44,48,59)(36,49)(37,42)(38,52)(39,43)(40,46,55,47)(41,50)(45,54,62,56)(51,57)(53,58)(60,63,64,61);(1,3)(2,7)(4,15)(5,19)(6,13)(8,24)(9,11)(10,27)(12,16)(14,21)(17,31)(18,23)(20,30)(22,26)(25,32)(28,29)(33,35)(34,40)(36,45)(37,47)(38,46)(39,48)(41,56)(42,59)(43,55)(44,52)(49,64)(50,63)(51,62)(53,61)(54,58)(57,60);(1,4,6,17)(2,8,9,25)(3,12,13,30)(5,20,21,16)(7,22,11,29)(10,28,18,26)(14,31,19,15)(23,32,27,24)(33,36)(34,41)(35,45)(37,50)(38,53)(39,51)(40,54)(42,57)(43,58)(44,60)(46,61)(47,63)(48,62)(49,52)(55,56)(59,64)

Polycyclic presentation:

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

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, 21417);