#6354250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354237
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1551 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)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,41)(30,38)(31,42)(33,39)(34,40)(35,46)(37,44)(43,47)(45,48);(17,19,22,32,23,31,21,30)(18,24,26,28,27,38,36,42)(20,29,35,44,34,45,33,43)(25,37,39,41,40,47,46,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,41)(30,43)(31,45)(32,44)(36,46)(38,47)(42,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,21,23,22)(18,26,27,36)(19,30,31,32)(20,33,34,35)(24,28,38,42)(25,39,40,46)(29,43,45,44)(37,41,47,48);(19,31)(20,34)(24,38)(25,40)(28,42)(30,32)(33,35)(39,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,e^c=eh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306706, #72306707, #72306742
maximal normal subgroup of	#72306706, #72306707, #72306742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261446);