#346506

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62866	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2204 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253702);