#347937

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
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

814 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)(3,10)(4,9)(5,7)(6,13)(8,11)(12,16)(14,15)(17,18,23,26)(19,22,29,40)(20,30,34,27)(21,35,39,25)(24,32,42,33)(28,38,36,37)(31,44,46,43)(41,45,48,47);(2,7)(8,15)(9,13)(10,16)(17,19,22,26,23,29,40,18)(20,27,32,42,34,30,33,24)(21,36,37,25,39,28,38,35)(31,44,45,41,46,43,47,48);(1,3)(2,8)(4,11)(5,12)(6,14)(7,15)(9,16)(10,13)(17,20)(18,24)(19,27)(21,31)(22,32)(23,34)(25,41)(26,42)(28,43)(29,30)(33,40)(35,48)(36,44)(37,45)(38,47)(39,46);(1,4,5,6)(2,9,7,13)(3,11,12,14)(8,16,15,10)(17,21,23,39)(18,25,26,35)(19,28,29,36)(20,31,34,46)(22,37,40,38)(24,41,42,48)(27,43,30,44)(32,45,33,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,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=dh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255133);