#494581

order	512
nilpotency class	4
isoclinism	isoclinic class of #33551
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1940 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fhi,d^c=di,e^c=eh,g^c=gh,f^e=fi,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401777);