#113755

order	512
nilpotency class	3
isoclinism	isoclinic class of #113742
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	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

720 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20951);