#436276

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

926 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)(7,15)(8,12,21,29)(9,22)(10,14)(11,17)(13,24)(16,31,33,27)(18,32)(19,36,34,25)(20,30,23,28)(26,35);(1,3,4,2)(5,7,9,17,10,18,24,35)(6,11,13,15,14,26,22,32)(8,19,20,33,21,34,23,16)(12,27,28,36,29,31,30,25);(2,3)(5,6)(7,11)(8,12)(9,13)(10,14)(15,17)(16,25)(18,26)(19,27)(20,28)(21,29)(22,24)(23,30)(31,34)(32,35)(33,36);(2,3)(5,8,9,23,10,21,24,20)(6,12,13,30,14,29,22,28)(7,16,17,34,18,33,35,19)(11,25,15,31,26,36,32,27)

Polycyclic presentation:

2^9|a^2=fg,b^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,h^d=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, 343472);