#436994

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

925 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)(5,6,11,16,10,14,27,15)(7,12,19,34,21,32,20,31)(8,22,28,26,24,9,23,13)(17,25,36,33,30,18,35,29);(5,7,11,19,10,21,27,20)(6,12,15,31,14,32,16,34)(8,17,23,35,24,30,28,36)(9,25,13,33,22,18,26,29);(5,8)(6,13)(7,17)(9,15)(10,24)(11,28)(12,29)(14,26)(16,22)(18,31)(19,36)(20,35)(21,30)(23,27)(25,34)(32,33);(2,4)(5,9,11,26,10,22,27,13)(6,8,15,28,14,24,16,23)(7,18,20,29,21,25,19,33)(12,30,34,36,32,17,31,35)

Polycyclic presentation:

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

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, 344190);