#6367641

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367534
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	#18619	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1476 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283574
maximal normal subgroup of	#72283574

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274837);