#6366605

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366542
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	#3106	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1473 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283751
maximal normal subgroup of	#72283751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273801);