#498893

order	512
nilpotency class	4
isoclinism	isoclinic class of #498890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2129 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 2129 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,h^b=hi,d^c=di,e^c=eh,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406089);