#6367115

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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	#18494	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1494 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)(9,13)(10,16)(11,14)(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);(1,2)(3,10)(4,13)(5,7)(6,9)(8,11)(12,15)(14,16)(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);(1,2)(3,8)(4,7)(5,13)(6,9)(10,11)(12,16)(14,15);(1,3,6,12)(2,8,9,16)(4,14,5,11)(7,15,13,10)(17,20,23,36)(18,25,28,43)(19,30,33,46)(21,35,22,37)(24,38,40,48)(26,42,27,44)(29,45,39,41)(31,34,32,47);(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=hi,d^2=h,e^2=hi,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=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274311);