#6364194

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
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

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283125
maximal normal subgroup of	#72283125

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271390);