#6817492

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	#3155	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgh,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,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, 6724688);