#421915

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	#23438	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329111);