#199574

order	512
nilpotency class	3
isoclinism	isoclinic class of #199522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106770);