#113214

order	512
nilpotency class	3
isoclinism	isoclinic class of #113206
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,f^c=fh,g^c=gh

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, 20410);