#115372

order	512
nilpotency class	3
isoclinism	isoclinic class of #115364
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3192	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

765 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,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=dh,e^b=ehi,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 22568);