#200375

order	512
nilpotency class	3
isoclinism	isoclinic class of #200152
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

992 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107571);