#343126

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

623 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:

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

Polycyclic presentation:

2^9|a^2=efi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250322);