#6355473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355448
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1608 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306853, #72306854, #72306925
maximal normal subgroup of	#72306853, #72306854, #72306925

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262669);