#6816698

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814819
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	#61289	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1536 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)(19,29)(20,27)(21,26)(22,25)(23,28)(24,32)(30,40)(31,45)(33,39)(34,48)(35,42)(36,41)(37,44)(38,47)(43,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,34)(21,32)(22,33)(23,30)(25,40)(26,29)(27,43)(28,39)(31,36)(35,46)(37,47)(38,41)(42,48)(44,45);(18,25,26,28)(19,30,32,33)(20,35)(27,44,42,41)(31,46,47,34)(36,37)(38,45)(43,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,26)(24,29)(25,28)(27,42)(38,45)(39,40)(41,44)(43,48);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,20,23,37,21,35,22,36)(18,27,28,44,26,42,25,41)(19,31,33,46,32,47,30,34)(24,38,40,48,29,45,39,43)

Polycyclic presentation:

2^9|c^2=gh,d^2=fi,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,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, 6723894);