#114558

order	512
nilpotency class	3
isoclinism	isoclinic class of #114551
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

727 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)(35,44,48,59)(36,41)(37,42)(38,53)(39,43)(40,46,55,47)(45,61,63,56)(49,64,62,54)(50,57)(51,60)(52,58);(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)(34,40)(36,49)(37,47)(38,46)(39,48)(41,56)(42,59)(43,55)(44,53)(45,51)(50,63)(52,62)(54,57)(58,61)(60,64);(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)(33,36,39,52)(34,41,43,58)(35,45,48,63)(37,51,38,50)(40,54,55,64)(42,60,53,57)(44,61,59,56)(46,49,47,62)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi,g^c=gh

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, 21754);