#321926

order	512
nilpotency class	3
isoclinism	isoclinic class of #321901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

682 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229122);