#289324

order	512
nilpotency class	3
isoclinism	isoclinic class of #289291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

547 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,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,g^b=ghi,f^c=fi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196520);