#498321

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405517);