#6355575

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60970	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6262771);