#435391

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
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

994 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)(5,6,11,30,12,17,10,16)(7,18,21,33,23,32,22,13)(8,24,20,14,26,34,25,31)(9,27,29,15,19,35,28,36);(5,7)(6,13,17,33)(8,9,26,19)(10,21)(11,22)(12,23)(14,27)(15,24)(16,18,30,32)(20,28,25,29)(31,35)(34,36);(5,8,11,20,12,26,10,25)(6,14,16,24,17,31,30,34)(7,19,22,29,23,9,21,28)(13,27,18,15,33,35,32,36);(2,4)(5,9)(6,15)(7,20)(8,21)(10,29)(11,28)(12,19)(13,31)(14,33)(16,35)(17,36)(18,34)(22,26)(23,25)(24,32)(27,30)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=f,e^2=i,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,g^b=gi,h^b=hi,e^c=ei,e^d=ei,g^d=gi,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, 342587);