#289993

order	512
nilpotency class	3
isoclinism	isoclinic class of #289947
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

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,7,10)(3,8,4,15)(5,14,16,13)(6,11,12,9)(17,18,21,26,22,27,20,25)(19,23,28,24,30,32,29,31);(1,3,12,14,7,4,6,13)(2,5,11,15,10,16,9,8)(17,19,20,29,22,30,21,28)(18,23,25,31,27,32,26,24);(1,4,7,3)(2,8,10,15)(5,9,16,11)(6,14,12,13)(17,19,22,30)(18,24,27,31)(20,28,21,29)(23,25,32,26);(1,5,6,8,7,16,12,15)(2,4,9,14,10,3,11,13)(17,20,22,21)(18,25,27,26)(19,28,30,29)(23,24,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197189);