#6815336

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814799
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	#61329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=f,e^2=h,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722532);