#237722

order	512
nilpotency class	3
isoclinism	isoclinic class of #237707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60917	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,d^c=dhi,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144918);