#6364557

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
prime radical	#62103	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1506 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=hi,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	#72306795, #72306805, #72306837
maximal normal subgroup of	#72306795, #72306805, #72306837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271753);