#348498

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	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

853 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,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, 255694);