#497892

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
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×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi

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, 405088);