#248378

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1005 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:

(1,2,7,12)(3,13,4,15)(5,11,6,10)(8,16,9,14)(17,18,22,26)(19,27,30,41)(20,31,33,42)(21,34,37,25)(23,35,38,29)(24,36,40,32)(28,39,45,47)(43,46,48,44);(1,3,7,4)(2,8,12,9)(5,16,6,14)(10,15,11,13)(17,19)(18,23)(20,28)(21,35)(22,30)(24,39)(25,41)(26,38)(27,34)(29,37)(31,43)(32,46)(33,45)(36,44)(40,47)(42,48);(1,3)(2,8)(4,7)(5,14)(6,16)(9,12)(10,13)(11,15)(23,38)(27,41)(39,47)(43,48);(1,4,7,3)(2,9,12,8)(5,14,6,16)(10,13,11,15)(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|a^2=h,b^2=i,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, 155574);