#6350551

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,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, 6257747);