#113226

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

769 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)(3,12,15,8)(4,16)(5,21)(6,11)(7,10)(9,18)(13,26,30,32)(14,24,22,25)(17,29,31,23)(19,27)(20,28)(33,34,38,43,39,44,37,42)(35,45,49,40,48,55,47,54)(36,41,50,56,51,57,52,58)(46,60,61,64,62,59,63,53);(1,3)(2,8)(4,17)(5,14)(6,22)(7,15)(9,26)(10,12)(11,24)(13,19)(16,23)(18,30)(20,31)(21,25)(27,32)(28,29)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,53)(42,55)(43,45)(44,54)(50,63)(51,62)(52,61)(56,59)(57,64)(58,60);(1,4,7,20)(2,9,10,27)(3,13,15,30)(5,19,6,18)(8,23,12,29)(11,28,21,16)(14,31,22,17)(24,26,25,32)(33,36)(34,41)(35,46)(37,52)(38,50)(39,51)(40,53)(42,58)(43,56)(44,57)(45,59)(47,63)(48,62)(49,61)(54,64)(55,60)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,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, 20422);