#114829

order	512
nilpotency class	3
isoclinism	isoclinic class of #114821
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,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, 22025);