#289140

order	512
nilpotency class	3
isoclinism	isoclinic class of #289115
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,g^b=gh,d^c=dh,f^c=fi,g^c=ghi

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, 196336);