#6369136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369070
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1626 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)(4,10)(5,9)(6,11)(8,12)(13,16)(14,15)(17,18,23,28)(19,29,33,46)(20,25,36,44)(21,27,22,26)(24,38,41,32)(30,40,34,48)(31,39,47,42)(35,45,37,43);(1,3)(2,7)(4,14)(5,12)(6,13)(8,9)(10,15)(11,16)(17,19,23,33)(18,24,28,41)(20,34,36,30)(21,32,22,38)(25,42,44,39)(26,29,27,46)(31,37,47,35)(40,45,48,43);(1,3)(2,8)(4,12)(5,14)(6,13)(7,9)(10,16)(11,15)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,39)(26,45)(27,43)(28,44)(29,40)(31,32)(33,34)(38,47)(41,42)(46,48);(1,3)(2,7)(4,14)(5,12)(6,13)(8,9)(10,15)(11,16)(17,20)(18,25)(19,31)(21,35)(22,37)(23,36)(24,40)(26,43)(27,45)(28,44)(29,39)(30,32)(33,47)(34,38)(41,48)(42,46);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,15,16,8)(17,21,23,22)(18,26,28,27)(19,32,33,38)(20,35,36,37)(24,29,41,46)(25,43,44,45)(30,47,34,31)(39,48,42,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306665, #72306670, #72306689
maximal normal subgroup of	#72306665, #72306670, #72306689

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276332);