#494623

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
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×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1908 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 401819);