#6356387

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356301
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60930	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1488 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263583);