#6816701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
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	C₂²×C₄	C₂⁵
prime radical	#61287	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18)(19,28)(20,24)(21,26)(22,27)(23,32)(25,35)(29,30)(31,43)(33,36)(34,44)(37,42)(38,45)(39,46)(40,47)(41,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,18)(19,29)(20,33)(21,25)(22,27)(23,32)(24,36)(26,35)(28,30)(31,40)(34,39)(37,41)(38,45)(42,48)(43,47)(44,46);(17,19,22,30)(18,23)(20,34,37,31)(21,32,35,45)(24,40)(25,28)(26,29)(27,38)(33,43)(36,47,48,46)(39,42)(41,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(19,30)(23,38)(28,29)(31,34)(32,45)(39,40)(43,44)(46,47);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,20,35,48,22,37,21,36)(18,24,25,41,27,42,26,33)(19,31,45,47,30,34,32,46)(23,39,28,43,38,40,29,44)

Polycyclic presentation:

2^9|c^2=gh,d^2=f,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

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, 6723897);