#200155

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

963 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)(3,11)(4,15,16,8)(5,18)(6,10)(7,13)(9,20)(12,25,27,32)(14,26)(17,23,19,24)(21,28,31,29)(22,30)(33,34)(35,40,46,56)(36,47,49,58)(37,51)(38,42)(39,43)(41,50,57,48)(44,59)(45,55,52,54)(53,62)(60,64)(61,63);(1,3,5,13,6,14,20,30)(2,7,9,11,10,22,18,26)(4,12,19,29,16,27,17,28)(8,21,24,25,15,31,23,32)(33,35)(34,40,43,56)(36,44)(37,45)(38,52)(39,46)(41,53,57,64)(42,54,51,55)(47,63,58,59)(48,60)(49,61)(50,62);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,24)(10,15)(11,25)(13,29)(14,27)(17,20)(18,23)(22,31)(26,32)(28,30)(33,36)(34,41)(35,44)(37,50)(38,48)(39,49)(40,53)(42,58)(43,57)(45,62)(46,61)(47,51)(52,60)(54,59)(55,63)(56,64);(4,16)(8,15)(12,27)(17,19)(21,31)(23,24)(25,32)(28,29)(33,37,39,38)(34,42,43,51)(35,45,46,52)(36,48,49,50)(40,54,56,55)(41,47,57,58)(44,60,61,62)(53,63,64,59)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,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, 107351);