#347900

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

803 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,23,27)(19,28,26,22)(20,24,33,38)(21,25,35,40)(29,41,37,32)(30,42,39,34)(31,44,46,36)(43,45,47,48);(2,6)(7,15)(8,13)(9,16)(17,19,22,18,23,26,28,27)(20,29,32,24,33,37,41,38)(21,30,34,25,35,39,42,40)(31,43,45,36,46,47,48,44);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20)(18,24)(19,29)(21,31)(22,32)(23,33)(25,36)(26,37)(27,38)(28,41)(30,43)(34,45)(35,46)(39,47)(40,44)(42,48);(17,21)(18,25)(19,30)(20,31)(22,34)(23,35)(24,36)(26,39)(27,40)(28,42)(29,43)(32,45)(33,46)(37,47)(38,44)(41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,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, 255096);