#148452

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1817
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 182 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55648);