#348492

order	512
nilpotency class	3
isoclinism	isoclinic class of #348478
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

743 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255688);