#348501

order	512
nilpotency class	3
isoclinism	isoclinic class of #348478
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	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 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:

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

Polycyclic presentation:

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

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, 255697);