#497764

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,g^b=gi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404960);