#7623663

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623662
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1899 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,g^a=gi,h^a=hi,d^b=df,f^b=fh,h^b=hi,g^c=gi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285648
maximal normal subgroup of	#72285648
maximal quotient of	#72342594, #72342595, SD₃₂×AGL(1,ℤ/8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530859);