#147533

order	512
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1026 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 206 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:

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

Polycyclic presentation:

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

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, 54729);