#171008

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 78204);