#436374

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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)(5,6,10,14,11,16,9,15)(7,17,20,12,21,28,19,29)(8,22,26,13,24,32,25,31)(18,33,34,30,35,27,23,36);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,28)(15,17)(16,29)(18,26)(22,33)(24,34)(25,35)(27,32)(31,36);(5,8,9,25,11,24,10,26)(6,13,14,32,16,31,15,22)(7,18,20,34,21,35,19,23)(12,27,17,30,29,33,28,36);(2,4)(7,19,21,20)(8,24)(12,28,29,17)(13,31)(18,34,35,23)(22,32)(25,26)(27,30,33,36)

Polycyclic presentation:

2^9|a^2=efg,c^2=e,d^2=ef,e^2=i,f^2=i,g^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,g^b=gi,h^b=hi,d^c=di

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, 343570);