#148179

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1387 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1387 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,f^c=fi,h^c=hi,f^e=fi

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, 55375);