#6357005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1524 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)(3,5)(4,11)(6,9)(7,10)(8,14)(12,15)(13,16)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,32)(29,44)(30,45)(31,38)(33,48)(34,41)(35,43)(36,42)(39,46)(40,47);(1,3,5,2,7,9,6,10)(4,12,13,16,15,11,14,8)(17,19,22,32,23,29,21,31)(18,24,26,28,27,38,37,44)(20,33,34,47,35,30,36,46)(25,40,42,48,43,39,41,45);(1,4,7,15)(2,8,10,16)(3,11,9,12)(5,13,6,14)(19,29)(24,38)(28,44)(30,33)(31,32)(39,40)(45,48)(46,47);(1,5,7,6)(2,3,10,9)(4,13,15,14)(8,11,16,12)(17,20,23,35)(18,25,27,43)(19,30,29,33)(21,36,22,34)(24,39,38,40)(26,42,37,41)(28,45,44,48)(31,47,32,46);(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,34,35,36)(24,28,38,44)(25,41,43,42)(30,46,33,47)(39,48,40,45)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283673
maximal normal subgroup of	#72283673

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264201);