#503871

order	512
nilpotency class	4
isoclinism	isoclinic class of #503870
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26130
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#444 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 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)(3,15)(4,22)(5,14)(6,12)(7,30)(8,19)(9,24)(10,23)(11,16)(13,29)(17,45)(18,34)(20,43)(21,36)(25,39)(26,40)(27,37)(28,42)(31,61)(32,41)(33,50)(35,49)(38,54)(44,48)(46,53)(47,56)(51,62)(52,64)(55,63)(57,59)(58,60);(1,3,16,19,6,21,5,27)(2,8,11,36,12,37,14,15)(4,18,23,54,24,55,26,53)(7,31,48,56,29,49,28,20)(9,38,40,63,22,46,10,34)(13,43,42,61,30,47,44,35)(17,32,51,25,52,58,50,59)(33,41,45,39,62,60,64,57);(1,4)(2,9)(3,17,21,52)(5,23)(6,24)(7,32)(8,33,37,62)(10,11)(12,22)(13,41)(14,40)(15,45,36,64)(16,26)(18,49,55,31)(19,50,27,51)(20,54,56,53)(25,28)(29,58)(30,60)(34,35,63,61)(38,47,46,43)(39,44)(42,57)(48,59);(1,4)(2,10)(3,18)(5,26)(6,24)(7,25)(8,34)(9,11)(12,40)(13,41)(14,22)(15,46)(16,23)(17,49)(19,54)(20,51)(21,55)(27,53)(28,32)(29,59)(30,60)(31,52)(33,35)(36,38)(37,63)(39,42)(43,45)(44,57)(47,64)(48,58)(50,56)(61,62)

Polycyclic presentation:

2^9|b^2=fh,c^2=fi,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411067);