#466411

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

911 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373607);