#249831

order	512
nilpotency class	3
isoclinism	isoclinic class of #249799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61800	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157027);