#147736

order	512
nilpotency class	4
isoclinism	isoclinic class of #129431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 142 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)(7,15,18,29)(8,19,21,13)(9,23,24,12)(10,14)(11,16)(17,33)(20,27)(22,28)(25,35,26,32)(30,36)(31,34);(1,2)(3,4)(5,7,10,18)(6,11,14,27)(8,20,21,16)(9,17,24,34)(12,26,23,25)(13,29,19,15)(22,35,36,32)(28,33,30,31);(2,4)(6,12)(8,21)(11,25)(13,30)(14,23)(15,31)(16,20)(19,28)(22,36)(26,27)(29,33)(32,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=ghi,d^c=di,g^c=gh,g^d=gi,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, 54932);