#113671

order	512
nilpotency class	3
isoclinism	isoclinic class of #113669
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2815	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=ei,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, 20867);