#6364271

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364250
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

1688 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)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,23,28)(19,29,33,38)(20,25,22,27)(21,26,36,44)(24,32,40,30)(31,45,47,42)(34,41,37,43)(35,48,46,39);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19)(18,24)(20,30)(21,35)(22,32)(23,33)(25,38)(26,42)(27,29)(28,40)(31,34)(36,46)(37,47)(39,43)(41,48)(44,45);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,34,36,37)(24,38,40,29)(26,41,44,43)(31,46,47,35)(39,42,48,45);(17,21)(18,26)(19,31)(20,34)(22,37)(23,36)(24,39)(25,41)(27,43)(28,44)(29,45)(30,46)(32,35)(33,47)(38,42)(40,48);(3,10)(6,14)(11,16)(13,15)(17,20,23,22)(18,27,28,25)(19,32,33,30)(21,34,36,37)(24,38,40,29)(26,43,44,41)(31,35,47,46)(39,42,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306963, #72307011, #72307031
maximal normal subgroup of	#72306963, #72307011, #72307031

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271467);