#113230

order	512
nilpotency class	3
isoclinism	isoclinic class of #113215
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20426);