#6352574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352469
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(2,8)(5,6)(9,16)(10,15)(11,14)(12,13)(17,18,22,25)(19,27,30,40)(20,24,31,38)(21,33,35,26)(23,34,37,29)(28,41,42,48)(32,45,44,39)(36,46,47,43);(1,2,7,11)(3,9,4,10)(5,8,6,14)(12,15,13,16)(17,19,22,30)(18,23,25,37)(20,28,31,42)(21,34,35,29)(24,36,38,47)(26,40,33,27)(32,46,44,43)(39,48,45,41);(1,3)(2,9)(4,7)(5,12)(6,13)(8,15)(10,11)(14,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,36)(25,38)(26,39)(27,41)(29,43)(30,42)(33,45)(34,46)(35,44)(37,47)(40,48);(1,4,7,3)(2,10,11,9)(5,12,6,13)(8,15,14,16)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(41,48)(43,46);(1,5,7,6)(2,8,11,14)(3,12,4,13)(9,15,10,16)(18,25)(23,37)(24,38)(26,33)(27,40)(36,47)(39,45)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259770);