#253410

order	512
nilpotency class	3
isoclinism	isoclinic class of #253387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160606);