#450287

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
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	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

986 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357483);