#461726

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,18,11,17,12,16)(7,19,21,13,23,25,22,32)(8,14,27,20,28,34,26,33)(9,24,29,36,30,35,31,15);(5,7,10,21,11,23,12,22)(6,13,16,19,17,32,18,25)(8,24,26,15,28,35,27,36)(9,20,29,34,30,33,31,14);(5,8,12,26,11,28,10,27)(6,14,18,20,17,34,16,33)(7,15,21,24,23,36,22,35)(9,13,31,19,30,32,29,25);(2,4)(5,9,10,29,11,30,12,31)(6,15,16,35,17,36,18,24)(7,20,21,34,23,33,22,14)(8,25,26,13,28,19,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368922);