#273733

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62091	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

802 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,27,23,28,22,26)(19,29,38,41,32,40,31,24)(20,33,36,25,37,43,35,44)(30,45,46,42,47,39,34,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,23,32)(18,24,28,41)(20,34,37,46)(21,38,22,31)(25,42,44,48)(26,40,27,29)(30,35,47,36)(33,45,43,39);(2,7)(4,13)(6,14)(10,12)(17,20,23,37)(18,25,28,44)(19,30,32,47)(21,35,22,36)(24,39,41,45)(26,43,27,33)(29,42,40,48)(31,46,38,34);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,18)(19,24)(20,25)(21,27)(22,26)(23,28)(29,38)(30,39)(31,40)(32,41)(33,35)(34,42)(36,43)(37,44)(45,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180929);