#493782

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1100 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)(7,16)(8,21)(9,15)(10,25)(11,26)(12,17)(13,19)(14,18)(20,33)(22,36)(23,32)(24,28)(27,29)(30,34)(31,35);(5,7,10,19,12,20,9,11)(6,13,15,16,17,26,25,33)(8,18,23,34,24,35,22,27)(14,28,29,32,31,21,30,36);(5,8)(6,14)(7,18)(9,22)(10,23)(11,27)(12,24)(13,28)(15,29)(16,32)(17,31)(19,34)(20,35)(21,26)(25,30)(33,36);(5,9,12,10)(6,15,17,25)(7,11,20,19)(8,22,24,23)(13,16,26,33)(14,29,31,30)(18,27,35,34)(21,36,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400978);