#248374

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	#18547	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1051 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)(3,13)(4,8)(5,11)(6,10)(7,12)(9,14)(15,16)(17,18)(19,28)(20,24)(21,25)(22,27)(23,30)(26,35)(29,44)(31,42)(32,39)(33,41)(34,37)(36,47)(38,45)(40,48)(43,46);(1,3,7,14)(2,8,12,16)(4,5,15,6)(9,11,13,10)(17,19)(18,23)(20,29)(21,34)(22,31)(24,38)(25,28)(26,42)(27,37)(30,35)(32,47)(33,46)(36,41)(39,44)(40,43)(45,48);(1,4,7,15)(2,9,12,13)(3,5,14,6)(8,11,16,10)(17,20,22,33)(18,24,27,41)(19,29,31,46)(21,32,35,48)(23,36,37,38)(25,39,26,40)(28,43,42,44)(30,45,34,47);(1,4)(2,9)(3,5)(6,14)(7,15)(8,11)(10,16)(12,13)(18,25)(21,35)(23,28)(24,39)(26,27)(30,34)(32,48)(36,43)(37,42)(38,44)(40,41)(45,47)

Polycyclic presentation:

2^9|b^2=i,c^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, 155570);