#248379

order	512
nilpotency class	3
isoclinism	isoclinic class of #248359
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	#3109	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1031 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)(3,11)(4,9)(5,16)(6,13)(7,12)(8,14)(10,15)(17,18)(19,27)(20,31)(21,34)(22,26)(23,29)(24,32)(25,37)(28,39)(30,41)(33,42)(35,38)(36,40)(43,44)(45,47)(46,48);(1,3)(2,9)(4,7)(5,6)(8,15)(10,12)(11,13)(14,16)(17,19,22,30)(18,23,26,38)(20,28,33,45)(21,35,37,29)(24,39,40,47)(25,41,34,27)(31,43,42,48)(32,46,36,44);(1,4)(2,10)(3,7)(5,8)(6,15)(9,12)(11,14)(13,16)(23,38)(27,41)(39,47)(43,48);(1,5,8,4)(2,11,14,10)(3,6,15,7)(9,13,16,12)(17,20,22,33)(18,24,26,40)(19,28,30,45)(21,36,37,32)(23,39,38,47)(25,42,34,31)(27,43,41,48)(29,46,35,44)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155575);