#461653

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,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,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, 368849);