#6355559

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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1579 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,23,27)(19,28,31,38)(20,25,32,40)(21,26,22,36)(24,35,39,30)(29,42,43,48)(33,41,34,46)(37,45,47,44);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,19,21,35,23,31,22,30)(18,24,36,38,27,39,26,28)(20,29,33,45,32,43,34,44)(25,37,46,48,40,47,41,42);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(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);(3,10)(6,14)(11,16)(13,15)(20,32)(25,40)(29,43)(33,34)(37,47)(41,46)(42,48)(44,45);(2,8)(6,14)(7,9)(13,15)(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|a^2=h,b^2=fh,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.

Relationships

maximal subgroup of	#72306887, #72306888, #72306925
maximal normal subgroup of	#72306887, #72306888, #72306925

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262755);