#429697

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 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)(5,6,10,15)(7,17,20,12)(8,22)(9,14,26,21)(11,16,18,28)(13,27)(19,34)(23,35)(24,32)(25,33)(29,31)(30,36);(5,7,18,14,10,20,11,21)(6,12,28,9,15,17,16,26)(8,19,24,31,25,35,27,36)(13,30,22,34,32,29,33,23);(5,8)(6,13)(7,19)(9,23)(10,25)(11,27)(12,29)(14,31)(15,32)(16,33)(17,30)(18,24)(20,35)(21,36)(22,28)(26,34);(2,4)(5,9,18,17,10,26,11,12)(6,14,28,20,15,21,16,7)(8,23,24,30,25,34,27,29)(13,31,22,35,32,36,33,19)

Polycyclic presentation:

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

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, 336893);