#195788

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

977 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)(3,13,17,23)(4,18,21,25)(5,11)(6,10)(7,12)(8,15,24,16)(9,19,26,20)(14,28)(22,31)(27,30)(29,32)(33,34,38,43,39,42,37,51)(35,40,46,55,47,54,45,61)(36,48,49,41,50,57,52,56)(44,58,59,53,60,64,62,63);(1,3,5,16,7,17,6,15)(2,8,11,23,12,24,10,13)(4,14,19,30,21,31,20,29)(9,22,18,32,26,28,25,27)(33,35,38,46,39,47,37,45)(34,40,43,55,42,54,51,61)(36,44,49,59,50,60,52,62)(41,53,57,64,56,63,48,58);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,30)(17,31)(23,32)(24,28)(33,36)(34,41)(35,44)(37,52)(38,49)(39,50)(40,53)(42,56)(43,57)(45,62)(46,59)(47,60)(48,51)(54,63)(55,64)(58,61);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,25,26,18)(14,29,31,30)(22,32,28,27)(34,42)(40,54)(41,56)(43,51)(48,57)(53,63)(55,61)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102984);