#349173

order	512
nilpotency class	3
isoclinism	isoclinic class of #349150
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3480	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

781 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=di,f^c=fi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256369);